Flexural & Shear Stresses | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


 
 
 
Members Subjected to Flexural Loads 
Introduction: 
In many engineering structures members are required to resist forces that are applied laterally or 
transversely to their axes. These type of members are termed as beam. 
There are various ways to define the beams such as 
Definition I: A beam is a laterally loaded member, whose cross-sectional dimensions are small as 
compared to its length. 
Definition II: A beam is nothing simply a bar which is subjected to forces or couples that lie in a 
plane containing the longitudnal axis of the bar. The forces are understood to act perpendicular to the 
longitudnal axis of the bar. 
Definition III: A bar working under bending is generally termed as a beam. 
Materials for Beam: 
The beams may be made from several usable engineering materials such commonly among them are 
as follows: 
? Metal 
? Wood 
? Concrete 
? Plastic 
Examples of Beams: 
Refer to the figures shown 
below that illustrates the beam 
 
 
Fig 1                                                             Fig 2 
In the fig.1, an electric pole has been shown which is subject to forces occurring due to wind; hence it 
is an example of beam. 
Page 2


 
 
 
Members Subjected to Flexural Loads 
Introduction: 
In many engineering structures members are required to resist forces that are applied laterally or 
transversely to their axes. These type of members are termed as beam. 
There are various ways to define the beams such as 
Definition I: A beam is a laterally loaded member, whose cross-sectional dimensions are small as 
compared to its length. 
Definition II: A beam is nothing simply a bar which is subjected to forces or couples that lie in a 
plane containing the longitudnal axis of the bar. The forces are understood to act perpendicular to the 
longitudnal axis of the bar. 
Definition III: A bar working under bending is generally termed as a beam. 
Materials for Beam: 
The beams may be made from several usable engineering materials such commonly among them are 
as follows: 
? Metal 
? Wood 
? Concrete 
? Plastic 
Examples of Beams: 
Refer to the figures shown 
below that illustrates the beam 
 
 
Fig 1                                                             Fig 2 
In the fig.1, an electric pole has been shown which is subject to forces occurring due to wind; hence it 
is an example of beam. 
 
 
In the fig.2, the wings of an aeroplane may be regarded as a beam because here the aerodynamic 
action is responsible to provide lateral loading on the member. 
Geometric forms of Beams: 
The Area of X-section of the beam may take several forms some of them have been shown below: 
 
Issues Regarding Beam: 
Designer would be interested to know the answers to following issues while dealing with beams in 
practical engineering application 
•  At what load will it fail 
•  How much deflection occurs under the application of loads. 
Classification of Beams: 
Beams are classified on the basis of their geometry and the manner in which they are supported. 
Classification I: The classification based on the basis of geometry normally includes features such as 
the shape of the X-section and whether the beam is straight or curved. 
Classification II: Beams are classified into several groups, depending primarily on the kind of 
supports used. But it must be clearly understood why do we need supports. The supports are required 
to provide constrainment to the movement of the beams or simply the supports resists the movements 
either in particular direction or in rotational direction or both. As a consequence of this, the reaction 
comes into picture whereas to resist rotational movements the moment comes into picture. On the 
basis of the support, the beams may be classified as follows: 
Cantilever Beam: A beam which is supported on the fixed support is termed as a cantilever beam: 
Now let us understand the meaning of a fixed support. Such a support is obtained by building a beam 
into a brick wall, casting it into concrete or welding the end of the beam. Such a support provides both 
the translational and rotational constrainment to the beam, therefore the reaction as well as the 
moments appears, as shown in the figure below 
Page 3


 
 
 
Members Subjected to Flexural Loads 
Introduction: 
In many engineering structures members are required to resist forces that are applied laterally or 
transversely to their axes. These type of members are termed as beam. 
There are various ways to define the beams such as 
Definition I: A beam is a laterally loaded member, whose cross-sectional dimensions are small as 
compared to its length. 
Definition II: A beam is nothing simply a bar which is subjected to forces or couples that lie in a 
plane containing the longitudnal axis of the bar. The forces are understood to act perpendicular to the 
longitudnal axis of the bar. 
Definition III: A bar working under bending is generally termed as a beam. 
Materials for Beam: 
The beams may be made from several usable engineering materials such commonly among them are 
as follows: 
? Metal 
? Wood 
? Concrete 
? Plastic 
Examples of Beams: 
Refer to the figures shown 
below that illustrates the beam 
 
 
Fig 1                                                             Fig 2 
In the fig.1, an electric pole has been shown which is subject to forces occurring due to wind; hence it 
is an example of beam. 
 
 
In the fig.2, the wings of an aeroplane may be regarded as a beam because here the aerodynamic 
action is responsible to provide lateral loading on the member. 
Geometric forms of Beams: 
The Area of X-section of the beam may take several forms some of them have been shown below: 
 
Issues Regarding Beam: 
Designer would be interested to know the answers to following issues while dealing with beams in 
practical engineering application 
•  At what load will it fail 
•  How much deflection occurs under the application of loads. 
Classification of Beams: 
Beams are classified on the basis of their geometry and the manner in which they are supported. 
Classification I: The classification based on the basis of geometry normally includes features such as 
the shape of the X-section and whether the beam is straight or curved. 
Classification II: Beams are classified into several groups, depending primarily on the kind of 
supports used. But it must be clearly understood why do we need supports. The supports are required 
to provide constrainment to the movement of the beams or simply the supports resists the movements 
either in particular direction or in rotational direction or both. As a consequence of this, the reaction 
comes into picture whereas to resist rotational movements the moment comes into picture. On the 
basis of the support, the beams may be classified as follows: 
Cantilever Beam: A beam which is supported on the fixed support is termed as a cantilever beam: 
Now let us understand the meaning of a fixed support. Such a support is obtained by building a beam 
into a brick wall, casting it into concrete or welding the end of the beam. Such a support provides both 
the translational and rotational constrainment to the beam, therefore the reaction as well as the 
moments appears, as shown in the figure below 
 
 
 
Simply Supported Beam: The beams are said to be simply supported if their supports creates only 
the translational constraints. 
 
Some times the translational movement may be allowed in one direction with the help of rollers and 
can be represented like this 
 
Statically Determinate or Statically Indeterminate Beams: 
The beams can also be categorized as statically determinate or else it can be referred as statically 
indeterminate. If all the external forces and moments acting on it can be determined from the 
equilibrium conditions alone then. It would be referred as a statically determinate beam, whereas in 
the statically indeterminate beams one has to consider deformation i.e. deflections to solve the 
problem. 
Page 4


 
 
 
Members Subjected to Flexural Loads 
Introduction: 
In many engineering structures members are required to resist forces that are applied laterally or 
transversely to their axes. These type of members are termed as beam. 
There are various ways to define the beams such as 
Definition I: A beam is a laterally loaded member, whose cross-sectional dimensions are small as 
compared to its length. 
Definition II: A beam is nothing simply a bar which is subjected to forces or couples that lie in a 
plane containing the longitudnal axis of the bar. The forces are understood to act perpendicular to the 
longitudnal axis of the bar. 
Definition III: A bar working under bending is generally termed as a beam. 
Materials for Beam: 
The beams may be made from several usable engineering materials such commonly among them are 
as follows: 
? Metal 
? Wood 
? Concrete 
? Plastic 
Examples of Beams: 
Refer to the figures shown 
below that illustrates the beam 
 
 
Fig 1                                                             Fig 2 
In the fig.1, an electric pole has been shown which is subject to forces occurring due to wind; hence it 
is an example of beam. 
 
 
In the fig.2, the wings of an aeroplane may be regarded as a beam because here the aerodynamic 
action is responsible to provide lateral loading on the member. 
Geometric forms of Beams: 
The Area of X-section of the beam may take several forms some of them have been shown below: 
 
Issues Regarding Beam: 
Designer would be interested to know the answers to following issues while dealing with beams in 
practical engineering application 
•  At what load will it fail 
•  How much deflection occurs under the application of loads. 
Classification of Beams: 
Beams are classified on the basis of their geometry and the manner in which they are supported. 
Classification I: The classification based on the basis of geometry normally includes features such as 
the shape of the X-section and whether the beam is straight or curved. 
Classification II: Beams are classified into several groups, depending primarily on the kind of 
supports used. But it must be clearly understood why do we need supports. The supports are required 
to provide constrainment to the movement of the beams or simply the supports resists the movements 
either in particular direction or in rotational direction or both. As a consequence of this, the reaction 
comes into picture whereas to resist rotational movements the moment comes into picture. On the 
basis of the support, the beams may be classified as follows: 
Cantilever Beam: A beam which is supported on the fixed support is termed as a cantilever beam: 
Now let us understand the meaning of a fixed support. Such a support is obtained by building a beam 
into a brick wall, casting it into concrete or welding the end of the beam. Such a support provides both 
the translational and rotational constrainment to the beam, therefore the reaction as well as the 
moments appears, as shown in the figure below 
 
 
 
Simply Supported Beam: The beams are said to be simply supported if their supports creates only 
the translational constraints. 
 
Some times the translational movement may be allowed in one direction with the help of rollers and 
can be represented like this 
 
Statically Determinate or Statically Indeterminate Beams: 
The beams can also be categorized as statically determinate or else it can be referred as statically 
indeterminate. If all the external forces and moments acting on it can be determined from the 
equilibrium conditions alone then. It would be referred as a statically determinate beam, whereas in 
the statically indeterminate beams one has to consider deformation i.e. deflections to solve the 
problem. 
 
 
Types of loads acting on beams: 
A beam is normally horizontal where as the external loads acting on the beams is generally in the 
vertical directions. In order to study the behaviors of beams under flexural loads. It becomes pertinent 
that one must be familiar with the various types of loads acting on the beams as well as their physical 
manifestations. 
A. Concentrated Load: It is a kind of load which is considered to act at a point. By this we mean that 
the length of beam over which the force acts is so small in comparison to its total length that one can 
model the force as though applied at a point in two dimensional view of beam. Here in this case, force 
or load may be made to act on a beam by a hanger or though other means 
 
B. Distributed Load: The distributed load is a kind of load which is made to spread over a entire 
span of beam or over a particular portion of the beam in some specific manner 
 
In the above figure, the rate of loading ‘q' is a function of x i.e. span of the beam, hence this is a non 
uniformly distributed load. 
The rate of loading ‘q' over the length of the beam may be uniform over the entire span of beam, then 
we cell this as a uniformly distributed load (U.D.L). The U.D.L may be represented in either of the 
way on the beams 
 
Page 5


 
 
 
Members Subjected to Flexural Loads 
Introduction: 
In many engineering structures members are required to resist forces that are applied laterally or 
transversely to their axes. These type of members are termed as beam. 
There are various ways to define the beams such as 
Definition I: A beam is a laterally loaded member, whose cross-sectional dimensions are small as 
compared to its length. 
Definition II: A beam is nothing simply a bar which is subjected to forces or couples that lie in a 
plane containing the longitudnal axis of the bar. The forces are understood to act perpendicular to the 
longitudnal axis of the bar. 
Definition III: A bar working under bending is generally termed as a beam. 
Materials for Beam: 
The beams may be made from several usable engineering materials such commonly among them are 
as follows: 
? Metal 
? Wood 
? Concrete 
? Plastic 
Examples of Beams: 
Refer to the figures shown 
below that illustrates the beam 
 
 
Fig 1                                                             Fig 2 
In the fig.1, an electric pole has been shown which is subject to forces occurring due to wind; hence it 
is an example of beam. 
 
 
In the fig.2, the wings of an aeroplane may be regarded as a beam because here the aerodynamic 
action is responsible to provide lateral loading on the member. 
Geometric forms of Beams: 
The Area of X-section of the beam may take several forms some of them have been shown below: 
 
Issues Regarding Beam: 
Designer would be interested to know the answers to following issues while dealing with beams in 
practical engineering application 
•  At what load will it fail 
•  How much deflection occurs under the application of loads. 
Classification of Beams: 
Beams are classified on the basis of their geometry and the manner in which they are supported. 
Classification I: The classification based on the basis of geometry normally includes features such as 
the shape of the X-section and whether the beam is straight or curved. 
Classification II: Beams are classified into several groups, depending primarily on the kind of 
supports used. But it must be clearly understood why do we need supports. The supports are required 
to provide constrainment to the movement of the beams or simply the supports resists the movements 
either in particular direction or in rotational direction or both. As a consequence of this, the reaction 
comes into picture whereas to resist rotational movements the moment comes into picture. On the 
basis of the support, the beams may be classified as follows: 
Cantilever Beam: A beam which is supported on the fixed support is termed as a cantilever beam: 
Now let us understand the meaning of a fixed support. Such a support is obtained by building a beam 
into a brick wall, casting it into concrete or welding the end of the beam. Such a support provides both 
the translational and rotational constrainment to the beam, therefore the reaction as well as the 
moments appears, as shown in the figure below 
 
 
 
Simply Supported Beam: The beams are said to be simply supported if their supports creates only 
the translational constraints. 
 
Some times the translational movement may be allowed in one direction with the help of rollers and 
can be represented like this 
 
Statically Determinate or Statically Indeterminate Beams: 
The beams can also be categorized as statically determinate or else it can be referred as statically 
indeterminate. If all the external forces and moments acting on it can be determined from the 
equilibrium conditions alone then. It would be referred as a statically determinate beam, whereas in 
the statically indeterminate beams one has to consider deformation i.e. deflections to solve the 
problem. 
 
 
Types of loads acting on beams: 
A beam is normally horizontal where as the external loads acting on the beams is generally in the 
vertical directions. In order to study the behaviors of beams under flexural loads. It becomes pertinent 
that one must be familiar with the various types of loads acting on the beams as well as their physical 
manifestations. 
A. Concentrated Load: It is a kind of load which is considered to act at a point. By this we mean that 
the length of beam over which the force acts is so small in comparison to its total length that one can 
model the force as though applied at a point in two dimensional view of beam. Here in this case, force 
or load may be made to act on a beam by a hanger or though other means 
 
B. Distributed Load: The distributed load is a kind of load which is made to spread over a entire 
span of beam or over a particular portion of the beam in some specific manner 
 
In the above figure, the rate of loading ‘q' is a function of x i.e. span of the beam, hence this is a non 
uniformly distributed load. 
The rate of loading ‘q' over the length of the beam may be uniform over the entire span of beam, then 
we cell this as a uniformly distributed load (U.D.L). The U.D.L may be represented in either of the 
way on the beams 
 
 
 
some times the load acting on the beams may be the uniformly varying as in the case of dams or on 
inclind wall of a vessel containing liquid, then this may be represented on the beam as below: 
 
The U.D.L can be easily realized by making idealization of the ware house load, where the bags of 
grains are placed over a beam. 
 
Concentrated Moment: 
The beam may be subjected to a concentrated moment essentially at a point. One of the possible 
arrangement for applying the moment is being shown in the figure below: 
 
Simple Bending Theory OR Theory of Flexure for Initially Straight Beams