In structural analysis, there are many methods to find deflection, rotation of beams, and frames this blog we will discuss the types of the method later we will elaborate on them.
A) for statically determinate beam
1:double integration method
2:M.H Macaulay's method
3: moment area method (MAM )
4: conjugate beam method
5:energy method's
6vereshagins method
B) for statically indeterminate beam:
1:force method/method of consistent deformation/ flexibility method
a) three-moment theorem: this is the part of the force method
2: displacement method :
a) slope displacement method
b) moment distribution method
c) stiffness matrix method
3: an iterative method
a)Kani's method
4: Approximate method:
a) portal frame method
b)cantilever method
c)point of inflection method
d) substitute frame method
e) column analogy method
but in this blog, we will only discuss the double integration method, M.H Macaulay's method, and the moment area method (MAM )
1)Double integration method :
In this method, we use integration to find deflection and rotation (displacements). The Equation used in this method is elastic curve equation; d2y/dx=- M/EI ---(1) with a certain limit
PROCEDURE:
a) find bending moment equation
b) Put the bending moment value in equation (1)
c)Integrate this and find dy/dx, which is nothing but the slope equation for beam---gives EQ (2)
d)Now again integrate the equation 2 and find y,y is the deflection equation for beams---gives EQ(3)
e)Now use the compatibility equation to find the integral constant c1 and c2.put these value in equation 2 and 3 and find slope and deflection
f) with the help of this equation we can find deflection and rotation at any point.
APPLICATION:
mostly used in statically determinate beams with symmetric loading with no eccentricity. Where the symmetry means the load is acting on the center of span see the figure for clarification
For example, a simply supported span is given with point load (p)at the center to find the deflection and rotation equation?
SOLUTION :
given figureSTEP 1: calculate the support reaction using the equilibrium equation.
equilibrium equation
(in our example there is no horizontal force )
∑fx= 0
∑fy= 0 (RA+RB = p)
∑mat any point = 0
(here we considerd point A)
∑mA=0
RB×l-P×l/2 =0
(clockwise moment is -ve and anticlockwise is +Ve)
RB=p/2
hence RA=RB=p/2
free body diagram (F.B.D)  |
| F.B.D |
STEP 2: cut the section x-x at the distance x from A to right and write bending moment equation about section xx
because of the symmetric loading, we will only solve the half portion other half potion is the same
 |
| at a distance, x cut the section |
now at section xx find moment mX= RA× x where 0<X≤l/2
STEP 3: put mx value in formula M=EI×d2y/dx
( because RA=p/2)
p/2× x =EI×d2y/dx
(integrate this value we know that dy/dx is slope and y is deflection)
EI∫d2y/dx =∫px/2
E1×dy/dx=px2/4 + C1........EQ(1)
again integrate equation (1)
E1×y=px3/12 + C1+ C2........EQ(2)
STEP 4: use the boundary condition in EQ(1) and EQ(2)
* at support, the deflection is always zero(means YA=0 At x=0 )
*at the simply supported beam with a point load at the center slope is zero at the center( means dy/dx=0 at x=l/2 )
E1×dy/dx=px2/4 + C1........EQ(1)
(put dy/dx=0 at x=l/2 in equation (1)and find C1)
EI×0=p(L/2)2×(1/4)+C1
C1=-pl2/16
E1×y=px3/12 + C1+ C2x........EQ(2)
( put YA=0 At x=0;C1=-pl2/16 in equation (2) and find C2 )
EI×0=p(0)3/12 -pl2/16+ C2
C2=0
now put C1and C2value in EQ(1) and EQ (2)
E1×dy/dx=px2/4 -pl2/16........EQ(3)
E1×y=px3/12-pl2/16x+ C2........EQ(4)
now equation 3 is called slope equation and equation 4 is called deflection equation
put any value of x(from 0 to l/2) and find deflection and slop at that point
2)M.H Macaulay's method :
This method is nothing but the ungraded version of the double integration method. In this method d2y/dx=- M/EI this equation is the same
PROCEDURE:
the procedure is the same as the double integration method but here we will write one equation for the whole span
APPLICATION:
Used for statically determinate beam but this method is also applicable for unsymmetric loading, eccentric loading unsymmetric loading means the load isn’t acting on the center of the span, and also we have to write bending moment equation for the entire span
Example:
3)Moment area method (MAM):
In this method, two theorems are used to determine the displacements (deflection and rotation)
a) theorem 1:
This theorem said that change in slope b/w two-point A and B (θab)is equal to the area of bending moment diagram divided by EI between these points A and B
θab= Area of bending moment diagram/EI
EI=flexural rigidity of span
E=youngs modulus
I=second moment of inertia of the section
b) theorem 2:
𝛿ab is the area of bending moment diagram between a and b multiplied by the C.G of the diagram from "A" divided by EI about
𝛿ba is the area of bending moment diagram between a and b multiplied by the C.G of the diagram from "A" divided by EI about
PROCEDURE:
1) if you want to find out the slope difference between two points A and B
a) draw the bending moment diagram between thee points
b) find the area of bending moment diagram between A and B, say Q is the area of bending moment diagram between A and B
c)divide the area by flexural rigidity E1 (Q/EI)
2) if you want to find out the deviation of A on tangent B
a) draw the bending moment diagram between thee points
b) find the area of bending moment diagram between A and B, say Q is the area of bending moment diagram between A and B
c) divide the area by flexural rigidity E1 (Q/EI)
d) now find C.G (center of gravity) of bending moment diagram about A Say X̄1
e)now multiply Q/ELwith X̄1
f) now δab=QX̄1 /EI is the deviation of A on tangent B
2) if you want to find out the deviation of B on tangent A
a) draw the bending moment diagram between thee points
b) find the area of bending moment diagram between A and B, say Q is the area of bending moment diagram between A and B
c) divide the area by flexural rigidity E1 (Q/EI)
d) now find C.G (center of gravity) of bending moment diagram about B Say X̄2
e)now multiply Q/ELwith X̄2
f) now δab=QX̄2/EI is the deviation of B on tangent A
in the further blog, I will give you some example regarding these methods SA1L01 this is the code for finding blog
SA1:means structural analysis 1
L means: lecture
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