# Methods of Frame Analysis

Elastic analysis of frames engages with the examination and study of the strength and properties of the members and structure at the working loads. Frames could be thoroughly examined with help of various techniques. Nonetheless, the techniques of examination adopted depends on the assortment of the frame, its pattern and arrangement (portal bay or multibay) multistoried frame and the extent of indeterminacy.

It is dependent on the following assumptions:

- The relation among the force and displacement is linear. (i.e. Hook’s law is utilized).
- Displacements are particularly minute in comparison to the geometry of the construction in the condition that they do not have any kind of affects on the analysis.

The techniques used for analysis of frame are:

- Flexibility coefficient technique.
- Slope displacement technique.
- Iterative techniques:

- Moment distribution technique(By Hardy Cross in 1930’s)
- Kani’s technique (by Gasper Kani in 1940’s)

- Approximate techniques:

- Substitute frame technique
- Portal technique
- Cantilever technique

**Flexibility Coefficient Technique:**

This technique is also known as the force technique or the compatibility technique. In this Redundant forces are taken into account as unknowns. Additional equations are derived by taking into account the geometrical conditions incurred on the formation of structures. This technique is utilizedd for the analyzing of the frames of lower D.O.R.

*Limitations:*

- This technique incorporates long computations even for very simple problems with small D.O.R.
- This technique becomes intractable for big D.O.R. (>3), when calculated manually particularly due to the simultaneous equations included.

This technique is not suitable for calculating, because a structure could be diminished to a statically determinate type in more than one technique.

**Slope Displacement Technique:**

It is stiffness technique or displacement or the equilibrium technique. It incorporates of the series of simultaneous equations, each one of them showing the relation between the moments acting at the ends of the members is expressed in forms of slope & deflection. The solution of slope deflection equations with the equilibrium equations promotes the values of unknown rotations of the joints. After knowing these rotations, the end moments are assessed with the help of the slope deflection equations.

Limitations:

- This technique is only useful in case for the structures/buildings with less Kinematic indeterminacy.
- The explanation of concurrent equation makes the technique monotonous for annual computations.

The formulation of the equilibrium circumstances tends to be a chief limitation in application of this technique.

**Approximate Techniques:**

Approximate investigation of hyper static structures gives a uncomplicated technique of acquiring speedy solutions for groundwork designs. It is a very helpful procedure that contributes to expand a appropriate design for final (meticulous) investigation of a structure, evaluate alternative designs & present a rapid check on the sufficiency of structural designs. These techniques utilize simplifying assumptions in regards to the structural property so as to acquire a quick answer to intricate structures. On the other hand, these techniqueologies have to be utilized with great care & not dependent upon for concluding designs, specifically in cases of the complex structures.

The regular procedure incorporates reducing the specified indeterminate arrangement to a structural system by bringing forward ample quantity of hinges. It is probable to test out the deflected profile of a structure/building for the specified loading & therefore position the points of inflection.

The analysis is undertaken independently for these two cases:

Vertical Loads:

The stress in the construction structure which is subjected to vertical loads is related to the relative rigidity of the beam & columns.

Horizontal Loads:

The property of the construction structure which is related to horizontal forces is based on its height to width proportion.

*3.1 Portal Technique:*

Because the shear deformations are most prevailing in low rise building structures, the technique makes abridged assumptions in regards to the horizontal shear in the columns. Each one of the bay of is subjected as a entrance frame, & horizontal force is dispersed among them in a equal manner.

*3.2 Cantilever Technique:*

This technique is basically utilized in case of the high rise structures. This is dependent on the simplifying assumptions in regards to the Axial Force in columns.

*3.3 Inflection Technique:*

The frame is diminished to a statically determinate shape by bringing forward sufficient number of points of the inflection. The loading on the frames frequently includes evenly dispersed dead loads & live loads.

*3.4 Substitute Frame Technique:*

The technique considers that the moments in the beams of the floor are manipulated by loading on that floor. The pressure of loading on the lower or upper floors is not taken into altogether. The procedure includes the division of multi-storied construction structure into the smaller frames. These sub frames are also called as equal frames or alternate frames.

**Iterative Technique:**

Iterative strategies shape an effective class of techniques for examination of uncertain structures. These strategies after exquisite and straightforward technique of investigation, that is sufficient for common structures.

These techniques depend on the conveyance of joint moments among members associated with a joint. The precision of the arrangement relies on the quantity of emphases performed; generally three or five cycles are satisfactory for the majority of the structures.

The moment distribution strategies were founded by Hardy Cross in 1930’s and by Gasper Kani in 1940’s. These techniques include dispersing the known settled moments of the auxiliary members to the neighboring members at the joints, so as to fulfill the states of the coherence of inclines and relocations.

In spite of the fact that these strategies are iterative in nature, they meet in a couple of cycles to give accurate solution.

*4.1 Moment Distribution Technique:*

This strategy was initially presented by Prof. Solid Cross is broadly utilized for the investigation of the intermediate structures. In this strategy first the basic framework is decreased to its kinematically determinate shape, this is achieved by accepting every one of the joints to be completely controlled. The settled end moments are ascertained for this state of structure. The joints are permitted to divert turn in a steady progression by discharging them progressively. The uneven moment at the joint shared by the imembers associated at the joint when it is discharged.

*4.2 Kant’s technique:*

This technique was presented by Gasper Kani in 1940’s. It includes circulating the obscure settled end moments of auxiliary members to adjoining joints, keeping in mind the end goal to fulfill the states of coherence of inclines and the displacements.

Process:

- Rotation stiffness at each one of the end of all the members of a construction structure is ascertained based on the end conditions.
- Both ends fixed

K_{ij}= K_{ji}= EI/L

- Near end fixed, far end simply supported

K_{ij}= ¾ EI/L; K_{ji}= 0

- Rotational factors are computed for all the members at each joint it is given by

U_{ij}= -0.5 (K_{ij}/ ?K_{ji})

{The sum of rotational factors at a joint is -0.5}

(Fixed end moments incorporating the transitional moments, moment releases and carry over moments are calculated for the members and entered. The total of the FEM at a joint is entered in the focal square drawn at the joint).

- Iterations could be started at any joint but the iterations start from the left end of the structure usually given by the equation

M?ij = Uij [(Mfi + M??i) + ? M?ji)]

- At first the rotational components? Mji (sum of the rotational moments at the far ends of the joint) could be taken for zero. Additional iterations consider the rotational moments of the preceding joints.
- Rotational moments are calculated at each joint consecutively till all the joints are processed. This process carries out one cycle of iteration.
- Steps 4 and 5 should be carried out repeatedly the difference in the values of rotation moments from consecutive cycles is neglected.
- Final moments in the members at each one of the joint are calculated from the rotational members of the final iterations step.

M_{ij} = (Mfij + M??ij) + 2 M?ij + M?jii

The lateral translation of joints (side sway) is taken into account by incorporating column shear in the iterative procedure.

- Displacement factors have to be calculated for each storey expressed by

Uij = -1.5 (Kij/?Kij)