Monday, 25 September 2017

Beam Deflections - Moment - Area Method

Moment - Area Method - Mohr's Theorems 

Otto Mohr (1835-1918)
 German Civil Engineer
In this method, to determining the slopes and deflections in beams involves the Area and Moment of the 'Bending moment diagram'. 
Deviation and Slope of Beam by Area-Moment Method
( B.M.D )
First Theorem :
The angle between tangents drawn at any two points on deflected curve, is equal to area of the M/EI diagram between the two points.
The angle between the tangents from points A and B 
O AB =  1/EI (Area of AB)

Second Theorem :
The intercept on a vertical line made by two tangents drawn at the two points on the deflected curve is equal to the moment of the M/EI diagram between two points about the vertical line.
The intercepts at point A,and B are,
t A/B = Z AB  =1/EI (Area of AB). XA
  and
t B/A BA  = 1/EI (Area of AB). XB
Sign Convention :
For +ve B.M. ,the area of  M/EI diagram is considered +ve , and for -ve B.M.,the area of M/EI diagram is considered as -ve  

  1. 1.Slope :
  1. Measured from left tangent, if θ is anti-clockwise, the change of slope is positive, negative if θ is clockwise.
  1. 2. Deflection
    1. The intercept  at any point is positive if it lies above the tangent, negative if the it is below the tangent.

    2. Application of Moment - Area Method for Cantilever Beams
        1.  

          General representation of deflection of cantilever beams

           

          From the figure above, the deflection at B denoted as δB is equal to the deviation of B from a tangent line through A denoted as tB/A. This is because the tangent line through A lies with the neutral axis of the beam.
          Similarly the angle between the tangents will equal to slope of the point. 
        2. Application of Moment - Area Method for Simply supported Beams


          The deflection δ at some point B of a simply supported beam can be obtained by the following steps:
          1.  

          Area-moment method of finding deflections in simply supported beam


          1. Compute the vertical intercept between Cand A  = ZCA
          2. Compute the vertical intercept between B and C  = ZBA
          3. Slope at support A = O A = t CA /L = ZCA / L
          4. Slope at support C = O C = tCA/ L = ZCA / L
    3.  Generally, the tangential deviation t is not equal to the beam deflection. In cantilever beams, however, the tangent drawn to the elastic curve at the wall is horizontal and coincidence therefore with the neutral axis of the beam. The tangential deviation or vertical intercept  in this case is equal to the deflection of the beam as shown below.

Tuesday, 12 September 2017

Deflection of Beams

Strength of Materials -1, Unit -5
The deformation of a beam is usually expressed in terms of its deflection from its original unloaded position. The deflection is measured from the original neutral surface of the beam to the neutral surface of the deformed beam. The configuration assumed by the deformed neutral surface is known as the elastic curve of the beam.

The angle through which the cross-section rotates with respect to the original position is called the angular rotation of the section.


Methods of Determining Beam Deflections:

1. Double integration method


The double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.

In Cartesian co-ordinates, the radius of curvature(R) of a curve y = f(x) is given by 

  In the derivation of flexure formula, the radius of curvature of a beam is given as

 1/R = M /EI

ThusThe deferential equitation of elastic curve is EI dy2 / dx2  = MX
The product EI is called the flexural rigidity of the beam, If EI is constant,the General equation may be written as:

 EI dy2 / dx2 =  MX     ---  (1)

Note :The down ward deflection will consider here as -ve ,
where x and y are the coordinates shown in the figure of the elastic curve of the beam under load, y is the deflection of the beam at any distance x. E is the modulus of elasticity of the beam, I represent the moment of inertia about the neutral axis, and M represents the bending moment at a distance x from the end of the beam. 
Direct Integration Method or Double Integration Method
The first integration of equation of (1) yields the slope of the elastic curve and the second integration equation (1) gives the deflection of the beam at any distance x.
The resulting solution must contain two constants of integration since EI y" = M is of second order.
These two Integral constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam.
BOUNDARY CONDITIONS:
1.A simply supported beam with rigid supports, 
at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero.
2.A Fixed support,
at fixed end, the deflection y is zero and slope dy/dx ( ø)is zero.
Macaulay's Method For Beam Deflections
For complex loading, specially where the span is partially loaded or loaded with number of concentrated loads,this Macaulay's method is more useful.
Note : 
1. While using the Macaulay's method, the section 'x' is to be taken in the last portion of the beam.
2. the quantity with in brackets ( ), should be integrated as a whole.
3.The expression for Bending Moment (Mx) equation can be used at any point, provided the term within the brackets becomes negative is omitted. 

Friday, 4 August 2017

BENDING STRESSES IN BEAMS - UINT 3

The stresses produced due to constant Bending Moment (with zero Shear Force or pure bending) are know as Bending stresses.

Assumptions in Theory of Bending :
In deriving the relations between the bending moments and flexure (bending)stresses and between the Shear forces and Sharing stresses the following assumptions are made.
1.Transverse sections of the beam that were plane before bending remain so even after bending.
2.The material of the beam is isotropic and homogeneous and follows Hooke's law and has the same value of Young's Modulus in tension and compression.
3.The beam is subjected to Pure bending and therefore bends in an arc of a circle.
4.The radius of curvature is large compared to the dimension of the cross-section.
5.Each layer is independent to enlarge or contract.
6.The stresses are purely longitudinal and local effects of point loads are neglected.

Flexure Formula :

Stresses caused by the bending moment are known as flexural or bending stresses. Consider a beam to be loaded as shown.

THE BENDING EQUATION: 

where R is the radius of curvature of the beam in mm, M is the bending moment in N·mm, fb is the flexural stress in MPa., I is the centroidal moment of inertia in mm4, and y is the distance from the neutral axis to the  fiber in mm.

THE BENDING EQUATION IS GIVEN BY, 

M/I = f/y = E/R 
Where 
M = Bending Moment
I = Moment of inertia about Neutral axis (N.A.)
f = Bending stress
y = Distance of the fiber from N.A.
R = Radius of Curvature
E = Young's Modulus
This equation may be remember as 

" May I flow you Every Rule "

M/I = f/y = E/R

Sunday, 30 July 2017

SHEAR FORCE - BENDING MOMENTS-1

Definition of a Beam

A beam is a bar subject to forces or couples that lie in a plane containing the longitudinal section of the bar. 
According to determinacy, a beam may be determinate or indeterminate.

Statically Determinate Beams

Statically determinate beams are those beams in which the reactions of the supports may be determined by the use of the three static equilibrium equations

Statically Indeterminate Beams
If the number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam.
The degree of indeterminacy is taken as the difference between the umber of reactions to the number of equations in static equilibrium that can be applied. 

Types of Supports

The supports of the beam may consists of 
a) Simply support or Roller Support
b) Hinge Support or Pinned Support
c) Fixed Support or Built in support

Types of Loading

Loads applied to the beam may consist of 
a) concentrated load (load applied at a point), 
b)uniform distributed load (u.d.l.), 
c) uniformly varying load, 
d) an applied couple or moment. 

Shear Force
The internal vertical resistance is called Shear Force.
 The Shear Force at a section may be defined as "the algebraic sum of the total vertical forces on either side of the cross section".
If a left portion of a section is considered ,upward forces will be positive and right side portion is considered down ward forces will be positive.

Bending moment
The moment witch bends the beam is called Bending moment.
The Bending moment at a section through a structural element may be defined as "the algebraic sum of the moments about that section of all external forces acting to one side of that section". 
The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium so the same bending moment will result from summing the moments, regardless of which side of the section is selected.

If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause "sagging", and a positive moment will cause "hogging"
It is more common to use the convention that a clockwise bending moment to the left of the point under consideration is taken as positive. 

Note:

1.Shear force at a section ,such that the portion to the right of the section slides up wards with respect to the left of the section are +ve and vice versa.

2.Bending moment at a section is +ve if it is sagging and -ve if it is hogging.

Shear Force and Bending moment diagrams 
Critical values within the beam are most commonly annotated using as a Shear force and bending moment diagrams, where negative values are plotted to scale below a horizontal line and positive values are plotted above the line.
A Shear Force diagram is one which shows the variation of the shear force along the length of the beam and a bending moment diagram is one which shows the variation of the bending moment along the length of the beam.
Bending moment  varies linearly over unloaded sections, and parabolically over uniformly loaded sections.
Point of contraflexure
A point of zero bending moment within a beam is called as point of contraflexure—that is the point of transition from hogging to sagging or vice versa.
Note:
1.Bending Moment is maximum at the point where the Shear Force is zero or where it changes direction from +ve to -ve or vice versa.
2.The point of contra-flexure or point of inflexion is the point where the Bending moment changes its direction from +ve to -ve or vice versa. Where the magnitude of B.M. is zero.
3. Relationship between intensity of load 'w' ,Shear Force 'F',and Bending Moment 'M'  :
    a) Rate of change of Shear Force is equal to Intensity of loading.
                          dF/dx = w
    b) Rate of change of Bending Moment is equal to Shear force.
                          dM/dx = F

S.M -- STRESS -STRAIN

 Elasticity is the property by virtue of which certain materials return back to their original position after the removal of the external force.
     Normal Stress or Direct Stress

The internal resistance per unit area, offered by a body against deformation is 
       known as   Normal stress.  
      The stress is given by   =R/A  =P/A.                                                               
      where P = External force or load;
                  A = Cross-sectional area.
   Stress is expressed as kgf/m2, kgf/cm2, N/m2 and N/mm2, or MPa

  1. The stress induced in a body, which is subjected to two equal and opposite pulls,      is known as  tensile stress
  2. The stress induced in a body, which is subjected to two equal and opposite pushes, is known as compressive stress.

Normal Strain
The deformation for unit length is called Strain.
   The ratio of change of dimension of the body to the original dimension is
       known as strain.
ROBERT HOOKE (1655-1705) 
Hooke’s law  : It states that within elastic limit,  the stress is proportional to the corresponding strain.

THOMAS YOUNG (1773-1829) 
 Young’s modulus: 
   The ratio of tensile stress (or compressive stress) to the corresponding strain is known as Young’s modulus or modulus of elasticity and is denoted by E.
The Normal Stress required to produce one unit normal strain with in elastic limit is also called Young's Modulus.

Rigidity Modulus:
The ratio of Shear Stress to the corresponding Shear Shear Strain with in the elastic limit is known as Modulus of Rigidity or Shear Modulus.
     It is denoted by 'C' or 'G'

11. The Total Change in length of a bar, when it subjected to an axial load of 'P' is
      dL = PL / AE
Factor of Safety.
12.The ratio between Ultimate stress to Working stress is known as Factor of Safety.

13. A composite bar is made up of two or more bars of equal lengths but of different materials rigidly fixed with each other and behaving as one unit for extension or compression.
14.The stresses induced in a body due to change in temperature are known as thermal stresses.
15. Thermal strain and thermal stress is given by
      thermal strain, e = α . T and
      thermal stress, p = α.T.E
      where       α = Co-efficient of linear expansion
                        T = Rise or fall of temperature
                        E = Young’s modulus

16. In case of a composite bar having two or more bars of different lengths, the extensions or compression in each bar will be equal. And the total load will be equal to the sum of the loads carried by each member.
17. In case of nut and bolt used on a tube with washers, the tensile load on the bolt is equal to the compressive load on the tube.
18. Elongation of a bar due to its own weight is given by
                   dL = W L / 2E
                       
      where       ω = Weight per unit volume of the bar material (W/L)
                         L = Length of bar.

Example :

For you to feel the situation, position yourself in pull-up exercise with your hands on the bar and your body hang freely above the ground. 
Notice that your arms suffer all your weight and your lower body fells no stress (center of weight is approximately just below the chest). 
If your body is the bar, the elongation will occur at the upper half of it.

Beam Deflections - Moment - Area Method

Moment - Area Method - Mohr's  Theorems  Otto Mohr (1835-1918)  German Civil Engineer In this method, to determining the slopes...