Slope deflection method of structural analysis

This is method of structural analysis which is given by the Prof. Maney in early of the 20th century (b/w 1910 to 1920). This is a displacement method of structural analysis, in which the end moment at the joints of the beams are expressed in the terms of the displacements and the fixed end moments. These expressions are known as the slope deflection equations.
Then the compatibility or the equilibrium equations are found for the structures, and these equilibrium equations are solved to get the unknown displacement at the joints. When the unknown displacements are found we can put these displacements in the slope deflection equations to find the final end moments. Once the final end moments are found, we can find the vertical reactions and the horizontal reactions at the various supports, to draw the shear force and the Bending moment diagrams.

A simple form of the slope deflection equation for a span AB of a continuous beam is written in the form as shown in the picture below.

In the beam shown in the figure, the kinematic indeterminacy is only two, i.e. angle at B and angle at C. So you need two equilibrium equations to get these values. Once you get the values, you can find the end moments by using the equations above.
The 2 equilibrium equations for the structure shown above can be created,on the basis of the fact that the sum of the left end moment and the right end moments at the joints B and C will be zero, because both of the joints are hinges, so they will not take any moment. So by equating the sum of the end moments at the two joints, and solving the two linear equations, simultaneously, I can get the unknown values of the slopes.
So now putting these values in the equations above, I can get the final end moments. 

Working stress method to design the concrete structures.

Working stress method of design is the old method which is replaced by the limit state design method now. It is still taught in the engineering studies, because it is important to understand the method, and some might also use this method.
In this method we take the design strength of the material as their yield strength. Further this strength is divided by a factor of safety to have a safer structure.
Generally the factor of safety for the concrete is almost equal to 3. So if you are using a M15 concrete then the design strength of the concrete in the bending compression for a RCC structure will be 5 N/sq.mm. M15 is not generally used for the construction of the RCC structures, generally M20 and M25 is used.
Design has two processes going side by side, one is analysis and other is the design. Structure is designed for the safety and economy at the same time, so we have to design the structural components by a number of trials until we get both economy and safety. 

Principle stresses and strains

There are generally two types of stresses, which we deal with in the structural mechanics or structural analysis of the Civil engineering. One is normal stress and another is shear stress. An element of a structure can be loaded with different combinations of the loading. There can be normal loading or shear loading on a face or the combination of both. These combination of loads results in the induction of the maximum stresses along some planes. There will be one particular plane of the element where the shear stress will be equal to zero but the normal stress will be maximum.
Principle Stress: 
This is the normal stress on a particular plane of a structural element subjected to external loads, where its value is maximum, but the value of the shear stress is zero.
Strain produced due to the principle stress is known as the principle strain. 

Strain energy stored in a member

When an elastic body is subjected to the external loading it may undergo the deformation or strain. If the material is strained withing its elastic limit, then according to the principle of conservation of energy, the work by the external load on the body should be equal to the strain energy that is stored in the body. Strain energy can also be called as the potential energy of the body.
If you remove the loading from the body, it will regain its original shape, so this energy is again converted to the mechanical energy.
Axial Loading on bar:
If a member having elasticity of "E" is subject to an axial loading of  stress "f",
then the amount of the strain energy stored in  body per unit of its length can be calculated to be equal to
U= f^2/2E
There are numbers of methods which uses the strain energy as the basis for the analysis of the given structures which can not be analysed by the general conditions of equilibrium, which are only 3 in numbers. Castigliano's theorems are much famous which use this energy principle also.
Bending loading on an elastic bar:
If the loading on the bar is a bending moment of value 'M', then the flexural stresses are induced in the cross section of the bar, which vary from the center of the cross section to the extreme fibers.
at any distance 'y' from the center, the value of stress is given by,
f = M.y/I
M is the applied moment, and I is the second moment of the area of the section about the central axis of the cross section.
Energy stored in any element of length, 'ds' and cross sectional area of 'da' is given by,
 (f^2.ds.da)/2E = (M.y/I)^2.ds.da/ 2E
Now the energy stored in the total cross section of the bar can be found by adding the energy stored in the infinite nos. of elements coming in the section of length ds.
now in the above equation, da.y^2 is the second moment of area. So summing up all the second moments of areas will give us the moment of Inertia of the section.
So the energy stored in the length of ds = M^2.ds/2EI
the total energy stored in the whole length of the bar can be found out by integrating the above equation with the limits of the length of the bar.

Indeterminacy (Determinate and indeterminate structures)

Hi,



There are the structures which can be easily analysed using the conditions of the equilibrium of the structure. These structures are easy to find their support reactions and the internal forces generated in the different members of these structures.

   

• Beams: 

The simply supported beam, beam with one end hinged one with roller support, perfect frames etc are the structures which are determinate. There are structures which have other support conditions, like if the beam has both ends fixed, or both ends hinged then the structure is statically indeterminate.
There are four reaction components if the beam is hinged at both the ends, and the conditions of the equilibrium are only three. So with these three conditions of equilibrium I can only find the 3 support reactions. So the structure is indeterminate.
if, R= total nos. of support reactions
    r = nos. of conditions of equilibrium.
then, the indeterminacy = R-r.
so in case of the fixed ends beam the indeterminacy is 1.

• Pinned frames:

In case of  frames, there are two kinds of indeterminacy. One is external and another is internal.
In case of the pinned frames, the external indeterminacy is R-r,
 but the internal indeterminacy is found by the formula:
I = m- (2j-r)
I = Internal indeterminacy
j= nos. of joints in the frame,
r= nos. of conditions of equilibrium.

• Rigid Frames:

In case rigid frames, the external indeterminacy is again R-r but the internal indeterminacy is equal to
I = 3a
a= nos. of the closed areas in the frame structure,
I = Internal indeterminacy.
Note: In any structure if a pin is introduced then it increases the numbers of equilibrium equations by 1. The equation is governed by the fact that the sum of all the moments is zero at the pin.

Thanks for your visit.