1) The document discusses using the moment area method to solve for the deflection of a statically indeterminate beam. 2) It provides the theorems of the moment area method and outlines the process of drawing the M/EI diagram, elastic curve, and using the theorems to calculate slope change and vertical deviation to determine deflection. 3) As an example, it then applies this method to find the maximum downward deflection of a small aluminum beam due to an applied force, showing the steps of determining the redundant reaction, drawing the moment diagram, M/EI diagram, elastic curve, and using the theorems to locate and calculate the maximum deflection.

1. Course No: CE 416
Course Title :Pre-stressed Concrete Lab
SOLVING STATICALLY
INDETERMINATE STRUCTURE BY
MOMENT AREA THEOREM
Submitted by
Submitted to
Name-Nabiha Nusrat
Munsi Galib Muktadir
ID no# 10.01.03.022
Lecturer,
&
Sabreena Nasrin
Lecturer,
Ahsanullah University of Science And Technology

2. INTRODUCTION
 There are many methods for solving indeterminate structures such as moment
distribution method, slope deflection method, stiffness method etc. Moment area
method is another one.
 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.

3. Scope of the study
 In numerous engineering applications where deflection of beams must be
determine, the loading is complex and cross sectional areas of the beam vary.
 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.

4. theorem
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

5. Theorem(Continue)
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. this
moment computed about point A where the deviation is to be determine.
Figure : Vertical deviation

6. 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.

7. process
Process to Draw M/EI diagram
1.
2.
Determine a redundant reaction, that establish the numerical values for the
bending moment diagram.
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.

8. 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.

9. 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.

10. 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.

11. is shown on the diagram below again.
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

12. ,
Problem(Continue)

13. Problem(Continue)

14. Problem(Continue)
The maximum deflection occurs where the tangent to the elastic curve is
horizontal, point C in the figure. Hence by noting that the tangent at A is also
horizontal and using the first moment theorem point C is located. When
hatched area in the figure having opposite signs are equal, that is, at a distance
2a = 2(4.2/56.8) = 0.148 m from A. The deviation
gives the deflection of
point C.

15. thank you