Theories of Elastic Failures

Theories of Failure:

There are some theories to predict the failure of materials at certain amount of applied load. Theories are checked experimentally and they are supported by the experimental results too.

(1) Maximum Principal stress Theory or Rankine's Theory:

If maximum principal stress is the design criteria then the maximum principal stress must not exceed the working stress for the material. For brittle materials, maximum principal theory is considered to be satisfactory because they do not fail by yield stress.

(2) Maximum Principal Strain Theory or St. Venant's Theory:

This theory says that a ductile material begins to yield when maximum principal strain reaches the strain at which yielding occurs in simple tension or when the minimum principal strain equals the yield point strain in simple compression.
This theory is satisfactory for the brittle materials but it has its drawbacks.

(3) Maximum Shear stress Theory or Guests theory:

The maximum shear stress is equal to the half of the difference between the maximum and minimum principal stresses. 

This theory gives fairly justified for ductile materials however doesn't give good results for the materials subjected to the hydro-static pressure. 

(4) Maximum strain energy theory or Haigh's theory:

This theory says that inelastic action at any point in a body due to any state of stress begins only when energy per unit volume absorbed at any point in a body due to any state of stress begins only when energy per unit volume absorbed at that point is equal to energy absorbed per unit volume of the material when subjected to elastic limit under a uni-axial state of stress as occurs in a simple tensile test. This theory has considerable experimental support for ductile material specially thick cylinders.

(5) Maximum shear strain energy theory or Mises Hensky theory:

The portion of the energy producing change in shape of the element is assumed to be completely responsible for the failure of material by yielding. This theory is in good agreement with experimental results of ductile materials.

(Reference: Made Easy, Strength of Materials by Er. R.K.Rajput)

Matrix Force Method of structural analysis

The following is a excerpt from a book which was published by the GK Publishers for the preparation of the GATE exam 2013. I have already written the introductory article to the matrix method, and you can get the understanding of it from my previous article of the same title. In this post I have taken the liberty to post the small content from the book, which will act as the advertisement of the book, so I hope everybody is happy with it.
Matrix method is the method of analysis of the indeterminate structures which uses the matrix approach to get the final moments at various joints in the structure.

Matrix Force and displacement Method: 
In case of analysis of beams by displacement method, joint rotations and displacements due to external forces are assumed as unknowns. Consider a continuous beam ABCD shown in figure(a) and (b) below:

The degree of the, NP, of the continuous beam is 3 that is three numbers of rotations can happen at the three joints B, C and D. The moments P1, P2 and P3 are the unbalanced fixed end moments due to external loads. X1, X2 and X3 are the rotations taken as unknowns.
At each joint, external forces P1, P2 and P3 are in equilibrium with the internal forces F1, F2, F3... F6.  Internal end rotations 'e' are the slopes of the tangents drawn to the elastic curve. Rotations are positive if they are clockwise and they are negative if the rotations are anticlockwise.  Similarly moments P1, P2 and P3 are positive if they are clockwise in direction and negative if anti clock wise.

Static Matrix A: Equilibrium condition between the external forces and the internal forces produces a static matrix.

Refer to the equilibrium diagram shown above,

P1= F2+F3,  P2 = F4+ F5  and P3= F6

                       In Matrix notations:

Here [A] is called a static matrix.

 Deformation Matrix B:

External joint forces P1, P2 and P3 cause joint rotations X1, X2 and X3 etc., these in turn cause internal joint rotations. Refer figure (b), where an external rotation X1= +1 will result in internal rotation e2=e3 = +1;                                                                                                                                                        In Matrix form :                                        Here [B] is called the deformation matrix.

Member stiffness Matrix:
Consider the figure given below:

Positive forces Fi and Fj are shown with the positive rotations ei and ej. Using the principle of conjugate beam it can be proved that,              
Fi = (4EI/l).ei  + (2EI/l).ej
Fj = (2EI/l). ei + (4EI/l). ej
Member stiffness matric 'S' shows the relationship between internal joint moments F1, F2 etc. in terms of joint rotations e1, e2 etc for the span AB, BC and CD.

Now , consider the following equation in the figure below:                                                                 4th equation is followed from 2nd and 3rd equation.
5th and 6th equation is followed from the 1st and 5th equation, first equation being the top most equation.
So the joint displacements X1, X2 are evaluated first from above equation and then joint forces F1, F2 are evaluated.
Final end moments are
F1f= Fixed End Moment1 +  F1
and F2f = Fixed End Moment2 + F2.

So this way you have found the value of the final end moments at the various joints.

Matrix approach to structural analysis

Matrix Method of Structural Analysis:
Matrix method of structural analysis is a matrix approach to solve the analysis problems of the structures. In this method the behavior of the structures written in the form of the forces and displacement equations, and these equations are then solved with matrix method of solving a system of simultaneous equations. This method is laborious when done manually but this has vast applications with the computers. So read on:
Here we divide the structure into a numbers of co-ordinates and then we define the forces and displacements at the co-ordinates in terms of their stiffness and their flexibility.
[P] = [k].[D]....(1)   Here, [k] is the stiffness matrix.
Above equation shows that we can write the force in terms of the stiffness of the joint and the amount of the displacement required. In this method, displacements are considered redundant and so this method is known as displacement method. You can find out the displacement by taking the inverse of the stiffness matrix on both sides and you can find out displacement matrix as the final result.
[D] = [f].[P]......(2)
here [f] is the flexibility matrix,
you can pre-multiply the above equation with inverse of flexibility matrix and you will get the force matrix. So this method is known as the force method. You have to treat the forces as the redundant and you can find them as the final result.
So in the force method and displacement method, the first step is to find the redundant forces and redundant displacements respectively. Then you have to form the flexibility matrix by producing unit force at the chosen co-ordinates. Then pre-multiply the inverse of the flexibility matrix on both sides of equation(2) and you will get the force matrix.
In displacement method, you have to form the stiffness matrix and pre-multiply its inverse on both sides of the equation(1), and you will get the displacements at the respective co-ordinates.
That is a small introduction to the matrix approach, if you want further assistance, please feel free to leave a comment.
Thank you,
:)

Muller Breslau Principle

Hi,
This principle is very much useful to determine the qualitative influence line for any function in a determinate or indeterminate beam.
Muller Breslau Principle : 
This principle states that the application of a function(reaction, shear or bending moment) deflects that beam in a shape which is same as the ILD of the given function at its point of application.
Let me explain with an example:

Reinforced concrete footings - Design

Few important things to remember about design of reinforced concrete footings:
(1)The thickness at the edge of a reinforced concrete footing resting on soil shall not be less than 15    cm.
(2) The thickness at the edge of a reinforced concrete footing on piles shall not be less than 30 cm.
(3) A foundation supporting of all the columns of a structure is called a raft footing.
(4) A common footing provided for two columns is called a combined footing.
(5) In the case of a common footing provided for two columns, where the projections beyond the edge column parallel to the length of the footing is restricted we must provide a rectangular combined footing.
(6) In case of a common footing provided for two columns, where the projection beyond the columns  parallel to the length of the footing are both restricted, we must provide a trapezoidal combined footing.
(7) A common foundation provided for a numbers of columns in a row is called a strip footing.
(8) For determining the shear force and bending moment for a footing the dead load of the footing is not considered when the footing rests on the soil.
(9) Weight of the footing may be assumed to be 10% of the column loads.
(10) For driving a pile into the ground a heavy hammer with a small drop should be used.
(11) For piles of length up to 30 times their least width the area of longitudinal reinforcement is not    less than 1.25% of the gross area.
(12) Piles of length more than 20 metres should be at least 45 cm* 45 cm.
(13) When a pile of length l is lifted at two points the distance of each support from the respective end should be  0.207*l.
(14)  A pile is lifted by two point suspension for the condition that the maximum bending moment is as small as possible. This maximum bending moment equals approximately  (w*l^2)/47.
(15) When a long pile of length l is lifted at three points, i.e. with one supporting point at the center and the other supporting points at a distance a from the respective end then, a=0.15*l.
(16) A pile of length l is lifted by three point suspension for the condition that the maximum bending  moment is as small as possible. This maximum bending moment is approximately  (w*l^2)/90.
(17) A pile of length l lying on the ground has to be hoisted to the vertical position. For the condition    that the maximum bending moment is as small as possible, the pile should be lifted at a point 0.293*l from the upper end.
(18) While calculating the ultimate bearing capacity of a pile by static formula the tip reaction may be taken equal to 1/5 of the ultimate load.
(19) A pile consisting of a vertical shaft of plain concrete with a bulb enlargement at the bottom is called a  pedestal pile.
(20)  A pile is driven into the soil by using a drop hammer. The best drop of the hammer is 120 cm.

Reference: Hand book of Civil Engineering by S.Ramamrutham.

Reinforced concrete columns - Design

Remember:

(1) The reinforced concrete column is a compression member the effective length of which exceeds three times the least lateral dimension.

(2) The reinforced concrete column is considered as a short column if its slenderness ratio l/b is less than 12.

(3) If in any plane one end of the column is unrestrained, its unsupported length l, shall not exceed 100*b*b/D.  where b= width of that cross section and D= depth of the cross section measured in the plane under consideration.

(4) All columns shall be designed for a minimum eccentricity of 20 mm.

(5)  All columns shall be designed for a minimum eccentricity equal to   l/500 + b/30  where l = unsupported length of the column and b= least lateral dimension of column.

(6) The area of the longitudinal reinforcement in a column shall not be less than 0.8% of the gross area.

(7) The area of longitudinal reinforcement in an R.C.C. column shall not exceed 6% of the gross area.

(8) The minimum numbers of longitudinal bars in a column of rectangular section is 4.

(9) The minimum numbers of longitudinal bars in a column of circular section is 6.

(10) The diameter of a bar of a column shall not be less than 12 mm.

(11) The spacing of longitudinal bars measured along the periphery of the column shall not exceed 300 mm.

(12) In a pedestal of a column the area of the longitudinal reinforcement shall not be less than 0.15% of the gross area.

(13) Pedestal is a compression member whose effective length does not exceed 3 times the least lateral dimension.

(14) When polygonal lateral ties are provided in a column the internal angles of the polygon shall not exceed 135 degrees.

(15)  The diameter of the lateral ties in a column shall not exceed the one-fourth of the diameter of the bars in longitudinal direction.

(16) The diameter of lateral ties in a column shall not be less than 5 mm.

(17) The pitch of the lateral ties shall not exceed 16 times the diameter of the longitudinal bars, and 48 times the diameter of the ties.

(18) The diameter of the helical reinforcement in a column shall not be less than 5 mm.

RCC Singly Reinforced Beams-working stress method

Hi, For design of the structures, we have different methods but here I am going to discuss here with you the working stress method of design of the singly reinforced beams.
Singly reinforced rectangular beams are the beams which are reinforced either at the top or at the bottom. Reinforcement is provided at top in case of the cantilever beams otherwise we provide the reinforcement generally at the bottom.

There are few assumptions in this method and some of the most important are:
All the tension is taken up by the reinforcement provided, so the tensile force of the concrete in the tensile section is ignored.

Critical Neutral Axis(Xc):  Location of the Critical neutral axis is found out using the geometrical relations between the stresses in the compression side and the tensile sides, which are assumed to vary linearly with the depth of the beam. Stresses are zero at the critical neutral axis and compressive on  one side and tensile on the other varying linearly. So there are two similar triangles formed and one can find out the location of the neautral axis using the relations between the various sides of the similar triangles.

Xc= mc/(t+mc) * d

here, Xc= Critical Neutral axis
m= modular ratio = 280/(3*compressive stress in concrete in bending)
c= maximum compressive stress in concrete
d= effective depth of the beam section

Actual Neutral axis(Xa):
Actual neutral axis of the section may be found by equating the moment of the areas on the two sides of the neutral axis about the neutral axis. On the compressive side the area of the concrete is considered  and on the tensile side the equivalent area of the steel reinforcement is taken and its moment is taken about the neutral axis. By equating the two we can get the actual neutral axis.

Balanced Section:
When Xa=Xc, it is a balanced section    (c'=c, ta=t)

Over-reinforced section:
When  Xa>Xc, it is a over- reinforced section   (c'= c, ta< t)

Under-reinforced section:
when Xa<Xc                                            (c'<c, ta=t)                                          

where,  (Xa= Actual neutral axis, Xc=Critical Neutral Axis
 c'- actual compressive stess in concrete, c- maximum compressive stress in concrete, ta= actual tensile stress in steel, t= maximum tensile stress in steel)
                 
Thanks for your visit!