Structural Analysis - I
Moment Area Method
The moment-area method,
developed by Otto Mohr in
1868 and later stated formally
by Charles E. Greene in 1873,
is a powerful tool for finding the
deflections of structures
primarily subjected to bending.
• The moment-area method is one of the most effective
methods for obtaining the bending displacement in beams
and frames.
• In this method, the area of the bending moment diagrams
is utilized for computing the slope and or deflections at
particular points along the axis of the beam or frame. Two
theorems known as the moment area theorems are utilized
for calculation of the deflection.
• One theorem is used to calculate the change in the slope
between two points on the elastic curve.
• The other theorem is used to compute the vertical distance
(called tangential deviation) between a point on the elastic
curve and a line tangent to the elastic curve at a second point.
• The method utilizes graphical interpretations of integrals
involved in the solution of the deflection differential equation
in terms of the areas and the moments of areas of the M/EI
diagram.
First Moment-Area Theorem
The change in slope between the tangents to the elastic curve at any two
points is equal to the area under the M/EI diagram between the two points,
provided that the elastic curve is continuous between the two points.
θA and θB are the slopes of the elastic curve at points A and B, respectively,
with respect to the axis of the beam in the undeformed (horizontal) state,
θBA denotes the angle between the tangents to the elastic curve at A and B.
In applying the first moment-area theorem, if the area of the M/EI diagram
between any two points is positive, then the angle from the tangent at the
point to the left to the tangent at the point to the right will be
counterclockwise, and this change in slope is considered to be positive;
and vice versa.
The deviation dt between the tangents drawn at the ends of the differential element dx
on a line perpendicular to the undeformed axis of the beam from a point B is given by
Integration of dt between points A and B yield the vertical distance tBA between the
point B and the tangent from point A on the elastic curve. Thus,
since the quantity M /EI represents an infinitesimal area under the M /EI diagram and
distance from that area to point B, the integral on right hand side can be interpreted as
moment of the area under the M/EI diagram between points A and B about point B
Second Moment-Area Theorem
The tangential deviation in the direction perpendicular to the undeformed
axis of the beam of a point on the elastic curve from the tangent to the
elastic curve at another point is equal to the moment of the area under the
M/EI diagram between the two points about the point at which the deviation
is desired, provided that the elastic curve is continuous between the two
points.
Note:
• Vertical intercept is not deflection – it is the distance from the deformed
position of the beam to the tangent of the deformed shape of the beam at
another location.
• It is important to note the order of the subscripts used. The first subscript
denotes the point where the deviation is determined and about which the
moments are evaluated, whereas the second subscript denotes the point
where the tangent to the elastic curve is drawn.
Example
Determine the end slope and deflection of the mid-point C in
the beam shown below using moment area method
Example
Determine the slopes and deflections at points B and C of the
cantilever beam shown below by the moment-area method.
Example
Use the moment-area method to determine the slopes at ends A
and D and the deflections at points B and C of the beam shown.
Example
Determine the maximum deflection for the beam shown by the
moment-area method.
Example
Use the moment-area method to determine the slope at point A
and the deflection at point C of the beam shown
Example
Determine the slope and deflection at the hinge of the beam
shown
