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Bending Moment
Bending Moment
Published: 2014/08/29
Channel: moodlemech
Internal Forces-Tension, Shear Force, Bending Moment
Internal Forces-Tension, Shear Force, Bending Moment
Published: 2012/02/24
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What is Concept of Shear Force and Bending Moment.
What is Concept of Shear Force and Bending Moment.
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Mechanical Engineering: Internal Forces on Beams (5 of 27) Bending Moments Explained
Mechanical Engineering: Internal Forces on Beams (5 of 27) Bending Moments Explained
Published: 2016/04/26
Channel: Michel van Biezen
Beams | Bending Moment and Shear Force Diagram
Beams | Bending Moment and Shear Force Diagram
Published: 2013/07/16
Channel: Learn Engineering
Shear, Bending Moment, Bending Stress Diagrams (Nazeer A. Khan)
Shear, Bending Moment, Bending Stress Diagrams (Nazeer A. Khan)
Published: 2012/05/20
Channel: Nazeer Khan
HSC Engineering Bending Moments and Shear Force Diagrams
HSC Engineering Bending Moments and Shear Force Diagrams
Published: 2012/11/16
Channel: HSCEngineering
Problem 1 on Shear Force and Bending Moment.
Problem 1 on Shear Force and Bending Moment.
Published: 2016/08/08
Channel: Ekeeda
Drawing Shear and Moment Diagrams Example- Mechanics of Materials and Statics
Drawing Shear and Moment Diagrams Example- Mechanics of Materials and Statics
Published: 2012/01/27
Channel: structurefree
05) Bending Moment (Elastic Case)
05) Bending Moment (Elastic Case)
Published: 2009/06/20
Channel: BioMechie
English - Finding Shear Force and Bending Moment Equations for a Simple Beam
English - Finding Shear Force and Bending Moment Equations for a Simple Beam
Published: 2012/01/06
Channel: CTSCIVIL
BENDING MOMENTS IN BEAMS - Simply supported beam with UDL
BENDING MOMENTS IN BEAMS - Simply supported beam with UDL
Published: 2013/11/14
Channel: Mike Bather
How to draw shear Force  & Bending moment diagram (Cantilever beam) -  GATE 2016 examination
How to draw shear Force & Bending moment diagram (Cantilever beam) - GATE 2016 examination
Published: 2015/12/02
Channel: CHINMAYACADEMY
Mechanics of Solids - IITM 2.9 Beams - BMD & SFD
Mechanics of Solids - IITM 2.9 Beams - BMD & SFD
Published: 2008/09/02
Channel: nptelhrd
How to Draw: SFD & BMD
How to Draw: SFD & BMD
Published: 2012/10/11
Channel: eLA Taylor's
Instructional Video: Bending Moment in a Beam Lab
Instructional Video: Bending Moment in a Beam Lab
Published: 2016/05/05
Channel: Cornelis Kok
Shear forces and bending moments: the basics
Shear forces and bending moments: the basics
Published: 2014/08/21
Channel: Structural Engineering Basics
EASY WAY TO DRAW SHEAR FORCE DIAGRAM AND BENDING MOMENT DIAGRAM
EASY WAY TO DRAW SHEAR FORCE DIAGRAM AND BENDING MOMENT DIAGRAM
Published: 2016/10/18
Channel: Short Notes
Part- 1 . Problem 1 Shear Force and Bending Moment.
Part- 1 . Problem 1 Shear Force and Bending Moment.
Published: 2015/11/01
Channel: Ekeeda
Shear Force and bending moment lecture in hindi , Strength of materials Part 5 TA0008
Shear Force and bending moment lecture in hindi , Strength of materials Part 5 TA0008
Published: 2016/10/01
Channel: Technical Atyachar
Bending Moment of a beam and Flexural Rigidity | BSc Physics in HINDI | EduPoint
Bending Moment of a beam and Flexural Rigidity | BSc Physics in HINDI | EduPoint
Published: 2016/12/01
Channel: EduPoint
Shear Force and Bending Moment Diagram (Gate lectures)
Shear Force and Bending Moment Diagram (Gate lectures)
Published: 2015/11/24
Channel: GateMentor
Statics Lecture 26: Internal forces -- Shear Force and Bending Moment Functions and Diagrams
Statics Lecture 26: Internal forces -- Shear Force and Bending Moment Functions and Diagrams
Published: 2013/03/15
Channel: Yiheng Wang
The basics of bending moment and shear force diagrams
The basics of bending moment and shear force diagrams
Published: 2012/11/02
Channel: Mark Evernden
Shear force and bending moment diagrams example #4: applied moment
Shear force and bending moment diagrams example #4: applied moment
Published: 2017/05/07
Channel: Engineer4Free
Lecture 13, Stress in beams subjected to bending moment and axial force (Lecture)
Lecture 13, Stress in beams subjected to bending moment and axial force (Lecture)
Published: 2016/10/10
Channel: Mechanics of Materials (Libre)
Structures III: Bending Moment Video
Structures III: Bending Moment Video
Published: 2008/09/11
Channel: MTI1550
Abhisek Arya- Shear force and bending moment diagrams
Abhisek Arya- Shear force and bending moment diagrams
Published: 2013/12/02
Channel: Jiet Jodhpur
Bending Moment Diagrams
Bending Moment Diagrams
Published: 2012/02/28
Channel: Darryl Morrell
bending moment model
bending moment model
Published: 2015/04/17
Channel: MrSgt911
Lecture on Shear Force & Bending Moment Diagram for Simply Supported Beams (In Hindi)
Lecture on Shear Force & Bending Moment Diagram for Simply Supported Beams (In Hindi)
Published: 2017/01/15
Channel: Mechanical Mentors by :- Ankur Malviya
Shear Force and Bending Moment Diagram | Strength of Materials | | GATE Preparation | ME
Shear Force and Bending Moment Diagram | Strength of Materials | | GATE Preparation | ME
Published: 2017/06/24
Channel: THE GATE ACADEMY
How to draw shear force & bending moment diagram (PART II) - GATE 2016 exam preparation
How to draw shear force & bending moment diagram (PART II) - GATE 2016 exam preparation
Published: 2015/10/08
Channel: CHINMAYACADEMY
Problem 1 on Shear Force Diagram and Bending Moment Diagram for simply supported beam
Problem 1 on Shear Force Diagram and Bending Moment Diagram for simply supported beam
Published: 2016/10/17
Channel: Ekeeda
SolidWorks beam bending shear and moment diagrams
SolidWorks beam bending shear and moment diagrams
Published: 2015/02/26
Channel: MCEN CU Boulder
shear force and bending moment diagram for simply supported beam with udl
shear force and bending moment diagram for simply supported beam with udl
Published: 2017/09/15
Channel: GTU MIMP
best method to draw shear force diagram(SFD) and bending moment diagram(BMD)
best method to draw shear force diagram(SFD) and bending moment diagram(BMD)
Published: 2017/06/03
Channel: VG academy Civil engineering
Shear Force and Bending Moment Diagram for Simply Supported Beam GATE in Hindi
Shear Force and Bending Moment Diagram for Simply Supported Beam GATE in Hindi
Published: 2016/03/27
Channel: Ujjwal Kumar Sen
GATE Lectures, Strength of Materials. Shear force and bending moment
GATE Lectures, Strength of Materials. Shear force and bending moment
Published: 2015/04/09
Channel: Ascent GATE Academy
How to draw shear force and bending moment diagram by Nooruddin Khan । Point load । Study Channel
How to draw shear force and bending moment diagram by Nooruddin Khan । Point load । Study Channel
Published: 2017/05/02
Channel: Study Channel
Bending moment and Shear force of beam calculation in Hindi by PARAG PAL
Bending moment and Shear force of beam calculation in Hindi by PARAG PAL
Published: 2016/08/17
Channel: parag pal
Problem 5 on Shear Force Diagram and Bending Moment Diagram for the Cantilever beam
Problem 5 on Shear Force Diagram and Bending Moment Diagram for the Cantilever beam
Published: 2016/10/17
Channel: Ekeeda
Part- 2 . Problem 1 Shear Force and Bending Moment.
Part- 2 . Problem 1 Shear Force and Bending Moment.
Published: 2015/11/01
Channel: Ekeeda
การเขียน Shear and Bending Moment Diagram ด้วยวิธี Inspection Method (ดร.ธเนศ วีระศิริ 1/2)
การเขียน Shear and Bending Moment Diagram ด้วยวิธี Inspection Method (ดร.ธเนศ วีระศิริ 1/2)
Published: 2016/07/11
Channel: tumcivil
CONCEPTS OF SHEAR FORCE AND BENDING MOMENT "STRENGTH OF MATERIAL"
CONCEPTS OF SHEAR FORCE AND BENDING MOMENT "STRENGTH OF MATERIAL"
Published: 2016/10/18
Channel: Short Notes
Problem  on Shear Force and Bending Moment for Cantilever Beam.
Problem on Shear Force and Bending Moment for Cantilever Beam.
Published: 2016/04/11
Channel: Ekeeda
BENDING MOMENTS IN BEAMS - Cantilever beam with UDL
BENDING MOMENTS IN BEAMS - Cantilever beam with UDL
Published: 2013/11/13
Channel: Mike Bather
[English - हिन्दी] - Shear Forces and Bending Moments Diagram : Part 1 (Basics)
[English - हिन्दी] - Shear Forces and Bending Moments Diagram : Part 1 (Basics)
Published: 2016/11/20
Channel: B.Tech Bro
Problem on Shear Force Diagram and Bending Moment Diagram.
Problem on Shear Force Diagram and Bending Moment Diagram.
Published: 2016/11/11
Channel: Ekeeda
SOM ( Shear force diagram and bending moment diagram of beams) Prof-Anup Goel-09325093084
SOM ( Shear force diagram and bending moment diagram of beams) Prof-Anup Goel-09325093084
Published: 2016/02/15
Channel: Anup Goel Engineering Study Center Pvt. Ltd.
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WIKIPEDIA ARTICLE

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Shear and moment diagram for a simply supported beam with a concentrated load at mid-span.(right)

A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend.[1][2] The most common or simplest structural element subjected to bending moments is the beam. The example shows a beam which is simply supported at both ends. Simply supported means that each end of the beam can rotate; therefore each end support has no bending moment. The ends can only react to the shear loads. Other beams can have both ends fixed; therefore each end support has both bending moment and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end (neither simple or fixed). In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.

The internal reaction loads in a cross-section of the structural element can be resolved into a resultant force and a resultant couple. For equilibrium, the moment created by external forces (and external moments) must be balanced by the couple induced by the internal loads. The resultant internal couple is called the bending moment while the resultant internal force is called the shear force (if it is transverse to the plane of element) or the normal force (if it is along the plane of the element).

The bending moment at a section through a structural element may be defined as "the sum of the moments about that section of all external forces acting to one side of that section". The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium so the same bending moment will result from summing the moments, regardless of which side of the section is selected. If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause "sagging", and a positive moment will cause "hogging". It is therefore clear that a point of zero bending moment within a beam is a point of contraflexure—that is the point of transition from hogging to sagging or vice versa.

Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres (N·m), or pound-foot (lbf·ft). The concept of bending moment is very important in engineering (particularly in civil and mechanical engineering) and physics.

Background[edit]

Tensile and compressive stresses increase proportionally with bending moment, but are also dependent on the second moment of area of the cross-section of a beam (that is, the shape of the cross-section, such as a circle, square or I-beam being common structural shapes). Failure in bending will occur when the bending moment is sufficient to induce tensile stresses greater than the yield stress of the material throughout the entire cross-section. In structural analysis, this bending failure is called a plastic hinge, since the full load carrying ability of the structural element is not reached until the full cross-section is past the yield stress. It is possible that failure of a structural element in shear may occur before failure in bending, however the mechanics of failure in shear and in bending are different.

Moments are calculated by multiplying the external vector forces (loads or reactions) by the vector distance at which they are applied. When analysing an entire element, it is sensible to calculate moments at both ends of the element, at the beginning, centre and end of any uniformly distributed loads, and directly underneath any point loads. Of course any "pin-joints" within a structure allow free rotation, and so zero moment occurs at these points as there is no way of transmitting turning forces from one side to the other.

It is more common to use the convention that a clockwise bending moment to the left of the point under consideration is taken as positive. This then corresponds to the second derivative of a function which, when positive, indicates a curvature that is 'lower at the centre' i.e. sagging. When defining moments and curvatures in this way calculus can be more readily used to find slopes and deflections.

Critical values within the beam are most commonly annotated using a bending moment diagram, where negative moments are plotted to scale above a horizontal line and positive below. Bending moment varies linearly over unloaded sections, and parabolically over uniformly loaded sections.

Engineering descriptions of the computation of bending moments can be confusing because of unexplained sign conventions and implicit assumptions. The descriptions below use vector mechanics to compute moments of force and bending moments in an attempt to explain, from first principles, why particular sign conventions are chosen.

Computing the moment of force[edit]

Computing the moment of force in a beam.

An important part of determining bending moments in practical problems is the computation of moments of force. Let be a force vector acting at a point A in a body. The moment of this force about a reference point (O) is defined as[2]

where is the moment vector and is the position vector from the reference point (O) to the point of application of the force (A). The symbol indicates the vector cross product. For many problems, it is more convenient to compute the moment of force about an axis that passes through the reference point O. If the unit vector along the axis is , the moment of force about the axis is defined as

where indicates the vector dot product.

Example[edit]

The adjacent figure shows a beam that is acted upon by a force . If the coordinate system is defined by the three unit vectors , we have the following

Therefore,

The moment about the axis is then

Sign conventions[edit]

The negative value suggests that a moment that tends to rotate a body clockwise around an axis should have a negative sign. However, the actual sign depends on the choice of the three axes . For instance, if we choose another right handed coordinate system with , we have

Then,

For this new choice of axes, a positive moment tends to rotate body clockwise around an axis.

Computing the bending moment[edit]

In a rigid body or in an unconstrained deformable body, the application of a moment of force causes a pure rotation. But if a deformable body is constrained, it develops internal forces in response to the external force so that equilibrium is maintained. An example is shown in the figure below. These internal forces will cause local deformations in the body.

For equilibrium, the sum of the internal force vectors is equal to the applied external force and the sum of the moment vectors created by the internal forces is equal to the moment of the external force. The internal force and moment vectors are oriented in such a way that the total force (internal + external) and moment (external + internal) of the system is zero. The internal moment vector is called the bending moment.[1]

Though bending moments have been used to determine the stress states in arbitrary shaped structures, the physical interpretation of the computed stresses is problematic. However, physical interpretations of bending moments in beams and plates have a straightforward interpretation as the stress resultants in a cross-section of the structural element. For example, in a beam in the figure, the bending moment vector due to stresses in the cross-section A perpendicular to the x-axis is given by

Expanding this expression we have,

We define the bending moment components as

The internal moments are computed about an origin that is at the neutral axis of the beam or plate and the integration is through the thickness ()

Example[edit]

Computing the bending moment in a beam.

In the beam shown in the adjacent figure, the external forces are the applied force at point A () and the reactions at the two support points O and B ( and ). The reactions can be computed using balances of forces and moments about point A, i.e.,

If is the length of the beam, we have

If we solve for the reactions we have

Looking at the free body diagram of the part of the beam to the left of point X, the total moment of the external forces about the point X is

If we compute the cross products, we have

For this situation, the only non-zero component of the bending moment is

For the sum of the moments at X about the axis to be zero, we require

At , we have .

Sign convention[edit]

In the above discussion, it is implicitly assumed that the bending moment is positive when the top of the beam is compressed. That can be seen if we consider a linear distribution of stress in the beam and find the resulting bending moment. Let the top of the beam be in compression with a stress and let the bottom of the beam have a stress . Then the stress distribution in the beam is . The bending moment due to these stresses is

where is the area moment of inertia of the cross-section of the beam. Therefore the bending moment is positive when the top of the beam is in compression.

Many authors follow a different convention in which the stress resultant is defined as

In that case, positive bending moments imply that the top of the beam is in tension. Of course, the definition of top depends on the coordinate system being used. In the examples above, the top is the location with the largest -coordinate.

See also[edit]

References[edit]

  1. ^ a b Gere, J.M.; Timoshenko, S.P. (1996), Mechanics of Materials:Forth edition, Nelson Engineering, ISBN 0534934293 
  2. ^ a b Beer, F.; Johnston, E.R. (1984), Vector mechanics for engineers: statics, McGraw Hill, pp. 62–76 

External links[edit]

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