Course No: CE 416 Course Title :Pre-stressed Concrete Lab

SOLVING STATICALLY INDETERMINATE STRUCTURE BY

MOMENT AREA THEOREM

Submitted byName-Nabiha NusratID no# 10.01.03.022

Submitted toMunsi Galib Muktadir Lecturer,&Sabreena NasrinLecturer,Ahsanullah University of Science And Technology

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.

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.

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

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

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.

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 supports2. 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.

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.

problemFind 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.

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.

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

is shown on the diagram below again.

Problem(Continue)

,

Problem(Continue)

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.

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