Thursday, December 27, 2012

Influence Line Diagrams

Influence Line Diagrams (ILD)
Influence line diagram is a diagram which represent the value of any particular function like Shear force, bending moment or axial force at particular point in the span of a structural member for the various positions of the point load on that structural member.
If you take a simply supported beam and you want to know the variation of the support reaction at one end with the variation of a particular point load from one end to the other end then drawing the Influence Line Drawing is the best option that you have.
Once you draw the ILD for the Reaction at one end, then you can find the value of the support reaction for any given point load at any particular position. It becomes very easy once the ILD is formed. 

Saturday, December 15, 2012

Hooke's Law

Strength of the material in measured or determined by performing some experiments on the material. The strength of a material is the maximum load per unit area which it can bear/withstand. 
When the tensile load is applied then the tensile strength of the material will resist the loading up to a limit and after that it will fail. The limit is known as the strength of that material.
Before getting failed the material gets deformed to certain degree and it depends on the Elastic properties of the material.  There is a famous law i .e. Hooke's law,  for the elastic materials, which helps us to determine the stress and strain in a material.
According to Hooke's law,
"Stress is directly proportional to strain, within elastic limits."
So if a material is stressed with an external load in its elastic range then the corresponding strain will follow this law.
Stress = Young's modulus of elasticity * Strain.

Young's modulus of elasticity is different for different materials and is same for all the elements formed of the same material.
Stress = Applied Load/ Area of application

and Strain = Deformation/ original dimension.

So strain do not have any units, only stress do have the units and that is N/mm2.

Saturday, December 8, 2012

Structural mechanics - Trusses -Perfect and Imperfect Frames

Trusses are the structural elements which are formed by joining different members or bars with the help of joints at their ends. Trusses are used to transfer the loads safely along  the girders/horizontal section on the portals to increase the stiffness.

Trussed bridge
 They can also be seen on the bridge portals or roofs etc. In actual practice members are generally welded/riveted to make the rigid joints and sometimes they may be bolted too.


To analyse a pin jointed frame we make some assumptions:

1) The joints are hinged(Pinned joints) so the net moment at any joint is zero.
2) The loads are applied only at the joints.
3) Only axial stresses are induced in the members.
4) Moments are the joints are zero.
5) Only axial forces(tensile and compressive) are induced in the member.

 In structural Engineering the frames are classified in two broad categories:
1) Perfect frames and    2)Imperfect frames.

Perfect frames are the frames which can be analysed to get the internal member forces and external support reactions by using the three conditions of static equilibrium. A simple example of a perfect frame is a triangular frame which is formed by joining the three member with the help of three joints.

Similarly Imperfect frames are the frames which can not be analysed to get the internal member forces and external support reactions by using the three conditions of static equilibrium.

Types of frames- Perfect and Imperfect(Deficient and redundant)

We know that in a triangular frames there are only three joints and there are three members.
For any structure to be in equilibrium the net force on the structure in the in either direction should be zero and also the net moment of all the forces at any joint should be equal to zero.

We can check it by taking the algebraic sum of all the forces in the two different directions for a plane frame(2-D structure) and it should be equal to zero.

Also take the moment of all the forces about a single joint and check if it is also equal to zero.So there are three conditions of equilibrium.
An indeterminate frame is a frame for which the three equations of the static equilibrium are not sufficient to analyze for its member forces and the external support reactions.

To check if the frame is perfect/determinate or imperfect remember the following:
For perfect frame:
m  = 2j -3
m=nos. of members in a frame
j = nos. of joints.
For a triangular frame m =3 and j = 3 so when we put these values in the equation above we get both the sides equal so, a triangular frame is a simple example of a perfect frame.

Now if m<2j-3 then the frame is called a deficient frame and the frame is unstable, i.e. the frame will deform if the external load is applied.

And if m>2j-3 then the frame is redundant. It is stable but we can not analyse the frame with the help of the conditions of the static equilibrium. and the difference between the m and 2j-3 is know as the redundancy of the frames.
For determinate frames/perfect frames, we can easily find out the external support reaction using the three equations of static equilibrium. To determine the forces in the members of the trusses,

There are two analytic methods to analyse the forces in the members of the trusses:

  1.  Method of Joints.
  2. Method of sections.

A) Method of Joints:

As per the assumptions made above the joints are hinged/pinned, and they loads are applied at the joints only. For the joints to be in static equilibrium, it must follow that the net force on a hinge must be zero.

In two dimensional frames/plane frames, we can further say,
1) Sum of all the horizontal forces is zero.
2) Sum of all the vertical forces is zero.

So if we follow the above two equations, we can find any two unknown forces in the members meeting at a joint. So to apply the method of joint one must remember the following statement:
Method of joint should be applied at the joints where the numbers of unknown member forces is two or less than two.

In a perfect frame to apply method of joints, you must start with a joint having two or less than two unknown member forces. Afterwards you can further move to the joints which earlier had more than two unknown member forces and now they will have two or less than two.


B) Method of Sections:

Method of section is based on the third equation of the static equilibrium discussed above, Sum of all the moments about a hinge is zero. We can analyze a section containing three or less than three unknown member forces, using the method of section.

When we cut a section(either to left or right) along three members, in perfect frames, two of them will meet at same joint.
Take the moment of all the forces about that joint and the moment of the two unknown forces will become zero(no perpendicular distance), so there will be the third unknown force in the equation of equilibrium.

Similarly, take the moment of all the forces at some other joint where other two forces(one previous one, and one just determined) meet. Automatically again there will be one unknown in the equation, so find that, and similarly you can go for third one.
Remember, for applying method of section, the section must contain three or less than three unknown member forces.

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Friday, December 7, 2012

Stress and Strain

Introduction: There are so many things around us and we do work with them to accomplish different goals.
We construct our houses and live in them because they provide us the protection from the harmful environment. For making houses we use different materials which are strong and durable to withstand their own weight and  surrounding environment.
For example we use the beams and columns to support the weight of the slab and the upper elements of the building. When the weight of the upper stories or the  slab is put on the beams and columns,they must not break or fail.When the external loads are applied on the column or beam the internal resistance is induced.
This internal resistance depends upon the physical and chemical composition of the material used in the manufacturing/construction of the beams and columns.
This internal resistance is denoted by a term 'stress'. Stress is a contact contact force, measured per unit area.
Stress = Contact force/Cross section area
The value of the internal resistance/stress is finite for every material. This finite limit of the internal resistance is known as the strength of the material.
The most important thing for a material to remain un-failed is that value of the internal resistance in the material must not reach the strength of the material when, they are applied with the external loads.

Now to  make it sure that the internal resistance do not reach the strength limit we have to calculate the magnitude of the internal stresses which gets induced in the structure due to the external loads.
After that we can decide the amount of the external load which can be put on the structural elements, safely. So there are three things which we must know to become a structure Engineer :
(1) strength of materials
(2) Theories of structure
(3) The design philosophy to design the structural elements for the various load requirements(Design) with different materials.

Stress :
When the external loads are applied on a structural elements the internal resisting stresses are developed in the elements.
When the axial loads are applied on the element (i.e. tensile or compressive) the axial/normal(tensile or compressive) stresses are induced.
So when the tensile force is applied tensile stresses gets induced and similarly when the compressive forces are applied the compressive stresses are induced.
These stresses are common for the  structural elements like columns.
Now when transverse loading is applied the induced stresses are called bending stresses. These stress are also of two types i.e. tensile and compressive.
Beams are the structural elements which are subjected to the transverse loading.
There is one more type of stress which is known as the shear stress. When the load is applied along the surface of the area of consideration such that it tries to shear the element into two parts along the surface then the resisting stresses are known as the shear stresses.

The simple formula to calculate the normal stress on a plane is = Applied Load/ Area of cross section

The units for the stress is N/mm2, KN/cm2 etc.

To be continued.......