Short Notes: Deflection of Beams | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Deflection of Beams 
The deformation of a beam is usually expressed in terms of its deflection from its 
original unloaded position. The deflection is measured from the original neutral 
surface of the beam to the neutral surface of the deformed beam. The 
configuration assumed by the deformed neutral surface is known as the elastic 
curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the 
actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on 
a loaded beam. There are many methods to find out the slope and deflection at a 
section in a loaded beam.
• The maximum deflection occurs where the slope is zero. The position of the 
maximum deflection is found out by equating the slope equation zero. Then 
the value of x is substituted in the deflection equation to calculate the 
maximum deflection
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These 
methods include:
Double Integration Method
This is most suitable when concentrated or udl over entire length is acting on 
the beam.A double integration method is a powerful tool in solving deflection
Page 2


Deflection of Beams 
The deformation of a beam is usually expressed in terms of its deflection from its 
original unloaded position. The deflection is measured from the original neutral 
surface of the beam to the neutral surface of the deformed beam. The 
configuration assumed by the deformed neutral surface is known as the elastic 
curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the 
actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on 
a loaded beam. There are many methods to find out the slope and deflection at a 
section in a loaded beam.
• The maximum deflection occurs where the slope is zero. The position of the 
maximum deflection is found out by equating the slope equation zero. Then 
the value of x is substituted in the deflection equation to calculate the 
maximum deflection
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These 
methods include:
Double Integration Method
This is most suitable when concentrated or udl over entire length is acting on 
the beam.A double integration method is a powerful tool in solving deflection
and slope of a beam at any point because we will be able to get the equation 
of the elastic curve.
• The double integration method is a powerful tool in solving deflection and 
slope of a beam at any point because we will be able to get the equation of
In calculus, the radius of curvature of a curve y = f(x) is given by
N N + -N (N -l)\ = -N { N -l)
• In the derivation of flexure formula, the radius of curvature of a beam is
p=EI/M
• Deflection of beams is so small, such that the slope of the elastic curve dy/dx 
is very small, and squaring this expression the value becomes practically 
negligible, hence
1 _ 1
P ~ tPy/dx2 ~ y"
Thus, E l/ M = 1 / y"
y
tt
m
• If El is constant, the equation may be written as:
Ely"=M
where x and y are the coordinates shown in the figure of the elastic curve of the 
beam under load,
Page 3


Deflection of Beams 
The deformation of a beam is usually expressed in terms of its deflection from its 
original unloaded position. The deflection is measured from the original neutral 
surface of the beam to the neutral surface of the deformed beam. The 
configuration assumed by the deformed neutral surface is known as the elastic 
curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the 
actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on 
a loaded beam. There are many methods to find out the slope and deflection at a 
section in a loaded beam.
• The maximum deflection occurs where the slope is zero. The position of the 
maximum deflection is found out by equating the slope equation zero. Then 
the value of x is substituted in the deflection equation to calculate the 
maximum deflection
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These 
methods include:
Double Integration Method
This is most suitable when concentrated or udl over entire length is acting on 
the beam.A double integration method is a powerful tool in solving deflection
and slope of a beam at any point because we will be able to get the equation 
of the elastic curve.
• The double integration method is a powerful tool in solving deflection and 
slope of a beam at any point because we will be able to get the equation of
In calculus, the radius of curvature of a curve y = f(x) is given by
N N + -N (N -l)\ = -N { N -l)
• In the derivation of flexure formula, the radius of curvature of a beam is
p=EI/M
• Deflection of beams is so small, such that the slope of the elastic curve dy/dx 
is very small, and squaring this expression the value becomes practically 
negligible, hence
1 _ 1
P ~ tPy/dx2 ~ y"
Thus, E l/ M = 1 / y"
y
tt
m
• If El is constant, the equation may be written as:
Ely"=M
where x and y are the coordinates shown in the figure of the elastic curve of the 
beam under load,
• y is the deflection of the beam at any distance x.
• E is the modulus of elasticity of the beam,
• I represent the moment of inertia about the neutral axis, and
• M represents the bending moment at a distance x from the end of the beam.
The product El is called the flexural rigidity of the beam.
Integrating one time
a & — [ m
dx J
The first integration y'(dy/dx) yields the Slope of the Elastic Curve
Second Integration
The second integration y gives the Deflection of the Beam at any distance x.
• The resulting solution must contain two constants of integration since El y" = 
M is of second order.
• These two constants must be evaluated from known conditions concerning 
the slope deflection at certain points of the beam.
• For instance, in the case of a simply supported beam with rigid supports, at x 
= 0 and x = L, the deflection y = 0, and in locating the point of maximum 
deflection, we simply set the slope of the elastic curve y' to zero
Area Moment Method
• Another method of determining the slopes and deflections in beams is the 
area-moment method, which involves the area of the moment diagram.The 
moment-area method is a
• The moment-area method is a semi graphical procedure that utilizes the 
properties of the area under the bending moment diagram. It is the quickest 
way to compute the deflection at a specific location if the bending moment 
diagram has a simple shape.
Page 4


Deflection of Beams 
The deformation of a beam is usually expressed in terms of its deflection from its 
original unloaded position. The deflection is measured from the original neutral 
surface of the beam to the neutral surface of the deformed beam. The 
configuration assumed by the deformed neutral surface is known as the elastic 
curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the 
actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on 
a loaded beam. There are many methods to find out the slope and deflection at a 
section in a loaded beam.
• The maximum deflection occurs where the slope is zero. The position of the 
maximum deflection is found out by equating the slope equation zero. Then 
the value of x is substituted in the deflection equation to calculate the 
maximum deflection
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These 
methods include:
Double Integration Method
This is most suitable when concentrated or udl over entire length is acting on 
the beam.A double integration method is a powerful tool in solving deflection
and slope of a beam at any point because we will be able to get the equation 
of the elastic curve.
• The double integration method is a powerful tool in solving deflection and 
slope of a beam at any point because we will be able to get the equation of
In calculus, the radius of curvature of a curve y = f(x) is given by
N N + -N (N -l)\ = -N { N -l)
• In the derivation of flexure formula, the radius of curvature of a beam is
p=EI/M
• Deflection of beams is so small, such that the slope of the elastic curve dy/dx 
is very small, and squaring this expression the value becomes practically 
negligible, hence
1 _ 1
P ~ tPy/dx2 ~ y"
Thus, E l/ M = 1 / y"
y
tt
m
• If El is constant, the equation may be written as:
Ely"=M
where x and y are the coordinates shown in the figure of the elastic curve of the 
beam under load,
• y is the deflection of the beam at any distance x.
• E is the modulus of elasticity of the beam,
• I represent the moment of inertia about the neutral axis, and
• M represents the bending moment at a distance x from the end of the beam.
The product El is called the flexural rigidity of the beam.
Integrating one time
a & — [ m
dx J
The first integration y'(dy/dx) yields the Slope of the Elastic Curve
Second Integration
The second integration y gives the Deflection of the Beam at any distance x.
• The resulting solution must contain two constants of integration since El y" = 
M is of second order.
• These two constants must be evaluated from known conditions concerning 
the slope deflection at certain points of the beam.
• For instance, in the case of a simply supported beam with rigid supports, at x 
= 0 and x = L, the deflection y = 0, and in locating the point of maximum 
deflection, we simply set the slope of the elastic curve y' to zero
Area Moment Method
• Another method of determining the slopes and deflections in beams is the 
area-moment method, which involves the area of the moment diagram.The 
moment-area method is a
• The moment-area method is a semi graphical procedure that utilizes the 
properties of the area under the bending moment diagram. It is the quickest 
way to compute the deflection at a specific location if the bending moment 
diagram has a simple shape.
Theorems of Area-Moment Method
• Theorem 1
° The change in slope between the tangents drawn to the elastic curve at 
any two points A and B is equal to the product of 1 /El multiplied by the 
area of the moment diagram between these two points
0A B =(1/EI)(AreaAB)
• Theorem 2
o The deviation of any point B relative to the tangent drawn to the elastic 
curve at any other point A, in a direction perpendicular to the original 
position of the beam, is equal to the product of 1 /El multiplied by the 
moment of an area about B of that part of the moment diagram between 
points A and B .
and
tB /A=(1/EI)(AreaAB)X-B 
W b=0 /EI)(AreaAB)X"A
Method of Superposition: The method of superposition, in which the applied 
loading is represented as a series of simple loads for which deflection formulas are 
available. Then the desired deflection is computed by adding the contributions of 
the component loads(principle of superposition). •
• Mostly direct formula is used in questions, hence it is advised to look for the 
beam deflection formula which are directly asked from this topic rather than 
going for long derivations.
Deflection for Common Loadings:
1. Concentrated load at the free end of cantilever beam
Page 5


Deflection of Beams 
The deformation of a beam is usually expressed in terms of its deflection from its 
original unloaded position. The deflection is measured from the original neutral 
surface of the beam to the neutral surface of the deformed beam. The 
configuration assumed by the deformed neutral surface is known as the elastic 
curve of the beam.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the 
actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on 
a loaded beam. There are many methods to find out the slope and deflection at a 
section in a loaded beam.
• The maximum deflection occurs where the slope is zero. The position of the 
maximum deflection is found out by equating the slope equation zero. Then 
the value of x is substituted in the deflection equation to calculate the 
maximum deflection
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These 
methods include:
Double Integration Method
This is most suitable when concentrated or udl over entire length is acting on 
the beam.A double integration method is a powerful tool in solving deflection
and slope of a beam at any point because we will be able to get the equation 
of the elastic curve.
• The double integration method is a powerful tool in solving deflection and 
slope of a beam at any point because we will be able to get the equation of
In calculus, the radius of curvature of a curve y = f(x) is given by
N N + -N (N -l)\ = -N { N -l)
• In the derivation of flexure formula, the radius of curvature of a beam is
p=EI/M
• Deflection of beams is so small, such that the slope of the elastic curve dy/dx 
is very small, and squaring this expression the value becomes practically 
negligible, hence
1 _ 1
P ~ tPy/dx2 ~ y"
Thus, E l/ M = 1 / y"
y
tt
m
• If El is constant, the equation may be written as:
Ely"=M
where x and y are the coordinates shown in the figure of the elastic curve of the 
beam under load,
• y is the deflection of the beam at any distance x.
• E is the modulus of elasticity of the beam,
• I represent the moment of inertia about the neutral axis, and
• M represents the bending moment at a distance x from the end of the beam.
The product El is called the flexural rigidity of the beam.
Integrating one time
a & — [ m
dx J
The first integration y'(dy/dx) yields the Slope of the Elastic Curve
Second Integration
The second integration y gives the Deflection of the Beam at any distance x.
• The resulting solution must contain two constants of integration since El y" = 
M is of second order.
• These two constants must be evaluated from known conditions concerning 
the slope deflection at certain points of the beam.
• For instance, in the case of a simply supported beam with rigid supports, at x 
= 0 and x = L, the deflection y = 0, and in locating the point of maximum 
deflection, we simply set the slope of the elastic curve y' to zero
Area Moment Method
• Another method of determining the slopes and deflections in beams is the 
area-moment method, which involves the area of the moment diagram.The 
moment-area method is a
• The moment-area method is a semi graphical procedure that utilizes the 
properties of the area under the bending moment diagram. It is the quickest 
way to compute the deflection at a specific location if the bending moment 
diagram has a simple shape.
Theorems of Area-Moment Method
• Theorem 1
° The change in slope between the tangents drawn to the elastic curve at 
any two points A and B is equal to the product of 1 /El multiplied by the 
area of the moment diagram between these two points
0A B =(1/EI)(AreaAB)
• Theorem 2
o The deviation of any point B relative to the tangent drawn to the elastic 
curve at any other point A, in a direction perpendicular to the original 
position of the beam, is equal to the product of 1 /El multiplied by the 
moment of an area about B of that part of the moment diagram between 
points A and B .
and
tB /A=(1/EI)(AreaAB)X-B 
W b=0 /EI)(AreaAB)X"A
Method of Superposition: The method of superposition, in which the applied 
loading is represented as a series of simple loads for which deflection formulas are 
available. Then the desired deflection is computed by adding the contributions of 
the component loads(principle of superposition). •
• Mostly direct formula is used in questions, hence it is advised to look for the 
beam deflection formula which are directly asked from this topic rather than 
going for long derivations.
Deflection for Common Loadings:
1. Concentrated load at the free end of cantilever beam
• Maximum Moment, M=-PL
• Slope at end, 0=PL2 /2EI
• Maximum deflection, 5=PL3 /3EI
• Deflection Equation (y is positive downward), Ely=(Px2 )(3L-x)/6 
2 .Concentrated load at any point on the span of cantilever beam
• Maximum Moment. M= -Pa
• Slope at end, 0=Pa2 /2EI
• Maximum deflection, 5=Pa3(3L-a)/6EI
• Deflection Equation (y is positive downward),
° Ely=Px2(3a-x)/6 for 0<x<a 
° Ely=Pa2(3x-a)/6 for a<x<L
3. Uniformly distributed load over the entire length of cantilever beam
y
• Maximum Moment, M=-w0L2 /2
• Slope at end, 0=woL3 /6EI
• Maximum deflection, 5=w0L4/8EI
• Deflection Equation (y is positive downward), Ely=wox2(6L2-4Lx+x2 )/120L
4. Triangular load, full at the fixed end and zero at the free end
y
Maximum Moment, M=-w0L2 / 6M