Statically indeterminate beam moment distribution methodTHANINCHANMALAI
The method of moment distribution is this:
Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the member for this condition.
At each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness“.
Multiply the moment distributed to each member at a joint by the carry-over factor at that end of the member and set this product at the other end of the member;
Distribute these moments just "carried over“.
Repeat the process until the moments to be carried over are small enough to be neglected.
Add all moments -- fixed-end moments, distributed moments, moments carried over -- at each end of each member to obtain the true moment at the end.
In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.
The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load, and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
Statically indeterminate beam moment distribution methodTHANINCHANMALAI
The method of moment distribution is this:
Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the member for this condition.
At each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness“.
Multiply the moment distributed to each member at a joint by the carry-over factor at that end of the member and set this product at the other end of the member;
Distribute these moments just "carried over“.
Repeat the process until the moments to be carried over are small enough to be neglected.
Add all moments -- fixed-end moments, distributed moments, moments carried over -- at each end of each member to obtain the true moment at the end.
In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.
The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load, and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
it contains the basic information about the shear force diagram which is the part of the Mechanics of solid. there many numerical solved and whivh will give you detaild idea in S.f.d.
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Moment area theorem presentation
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
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.