This document discusses solving statically indeterminate structures using the moment area method. It begins with an introduction to the moment area theorem developed by Otto Mohr and Charles Greene. The scope of the study is described as applicable to cantilever beams, simply supported beams with symmetrical loading, and beams fixed at both ends. The assumptions and two theorems of the moment area method are outlined. Theorem 1 relates the change in slope between two points to the area under the bending moment diagram between those points. Theorem 2 relates the vertical deviation of a tangent at one point from another to the moment of the area under the bending moment diagram between the points. An example problem is presented to demonstrate solving for the maximum downward deflection of a beam

1. Vishwakarma
Government Engineering
College, Chandkheda
Name:-Vekaria Darshil(150170106061)
Branch:-Civil
Sub:-SA-1 Sem:-4th
Topic:-SOLVING STATICALLY INDETERMINATE
STRUCTURES BY MOMENT AREA
METHOD.

2. INTRODUCTION:-
 The idea of moment area theorem was developed by Otto Mohr
and later started formally by Charles E. Greene in 1873.It is just an
alternative method for solving deflection problems.
 In this method we will establish a procedure that utilizes the area
of the moment diagrams [actually, the M/EI diagrams] to evaluate the
slope or deflection at selected points along the axis of a beam or
frame.
 In numerous engineering applications where deflection of beams
must be determine, the loading is complex and cross sectional areas
of the beam vary.

3. What is moment area method?
This is one of the easiest methods
for finding deflection and slope of a
beam with the help of area of
bending moment of the beam, which
is loaded with some load or any
moment at any point or on whole
span.

4. Scope of the study:-
When the superposition technique of indeterminate beam
accelerated according to following reasons restrained and continues
beams differ from the simply supported beams mainly by the
presence of redundant moment at the supports then moment area
method can be used.
It is convenient to use this method with great advantage in the
following type of problems:
1. Cantilever beams(slope at the fixed end is zero).
2. Simply supported beams carrying symmetrical loading.(slope at
mid span is zero)
3. Beams fixed at both ends(slope at each end is zero).

5. DEFLECTION OF BEAMS:-
Assumptions for slope and displacement
by MOMENT AREA THEOREM are as
follows:-
 Beam is initially straight.
Is elastically deformed by the loads, such
that the slope and deflection of the elastic
curve are very small.
 Deformations are caused by bending.

6. Theorem 1:-The change in slope between
any two points on the elastic curve equals the area of
the M/EI diagram between two points.
Figure : Interpretation of small change in an element

7. Theorem 2:- The vertical deviation of the tangent at
a point A on the elastic curve with respect to the tangent
extended from another B equals the moment of the area
under the M/EI diagram between the two points A and B.
Figure : Vertical deviation

8. Theorem(Continue):-
This method requires an accurate sketch of the deflected shape,
employs above two theorems. THEOREM 1 is used to calculate
a change in slope between two points on the elastic curve and
THEOREM 2 is used to compute the vertical distance (called a
tangential deviation) between
a point on the elastic curve
and a line tangent to the
elastic curve at a second
point.
Figure : Moment area theorem.

9. PROOF:-
------------(1)

10. For finding slope:-

11. For deflection:-

12. Now We get:-
Where M.ds=area of b.m. diagram
M.x.ds=moment of area of b.m. diagram

13. Process:-
Process to Draw M/EI diagram:-
1. Determine a redundant reaction, that establish the numerical values for
the bending moment diagram.
2. Divided moment diagram by EI. Plot the value and sketch the M/EI
Process to Draw Elastic Curve:-
1 Draw an exaggerated view of the beam’s curve. Recall that points of zero
slope occur at fixed supports and zero displacement occurs at all fixed, pin and
roller supports
2. If it becomes difficult to draw the general shape of the elastic curve, use the
M/EI diagram. Realize that when the beam is subjected to a positive moment
the beam bends concave up, where negative he negative moments bends the
beam concave down. And change in curvature occurs where the moment of the
beam is zero.

14. Process(continue):-
Process to Calculate Deviation:-
1. Apply theorem 1 to determine the angle between two tangents and
theorem 2 to determine vertical deviation between these tangents.
2. Realize that theorem 2 in general will not yield the displacement
of a point on the elastic curve. When applied properly it will only
give the vertical distance or deviation of a tangent at a point A on
the elastic curve from the tangent at B.
3. After applying either theorem 1 or theorem 2 the algebraic sign of
the answer can be verified from the angle or deviation as indicated
on the elastic curve.

15. Problem:-
Find the maximum downward deflection of the
small aluminum beam shown in figure due to an
applied force P=100N. The beam constant flexure
rigidity EI=60N.

16. Problem(Continue):-
Solution:The solution of this problem consists of
two parts. First a redundant reaction must be
determined to establish the numerical values for the
bending moment diagram. Then the usual moment-
area procedure is applied to find the deflection.

17. Problem(Continue):-
By assuming the beam is released from the redundant
end moment, a simple beam-moment diagram is
constructed as given here.
The moment diagram of known shape due to the
unknown redundant moment

18. Problem(Continue):-
,

19. Problem(Continue):-

20. Example:-

21. Diagrams:-
B.M. Diagram
M/EI diagram

22. Solution:-

23. THANK YOU