Elastic analysis deals with the study of strength and behavior of the members and structure at working loads. Frames can be analyzed by various methods. However, the method of analysis adopted depends upon the types of frame, its configuration (portal bay or multibay) multistoried frame and Degree of indeterminacy.

It is based on the following assumptions:

1. Relation between force and displacement is linear. (i.e. Hook’s law is applicable).

2. Displacements are extremely small compared to the geometry of the structure in the sense that they do not affect the analysis.

**The methods used for analysis of frame are:**

1. Flexibility coefficient method.

2. Slope displacement method.

3. Iterative methods like

a. Moment distribution method(By Hardy Cross in 1930’s)

b. Kani’s method (by Gasper Kani in 1940’s)

4. Approximate methods like

a. Substitute frame method

b. Portal method

c. Cantilever method

**1. FLEXIBILITY COEFFICIENT METHOD:**

This method is called as force method or compatibility method. In this Redundant forces are chosen as unknowns. Additional equations are obtained by considering the geometrical conditions imposed on the formation of structures. This method is used for analyzing frames of lower D.O.R.

· **Limitations:**

1. This method involves long computations even for simple problems with small D.O.R.

2. This method becomes intractable for large D.O.R. (>3), when computed manually especially because of simultaneous equations involved.

This method is not ideal for computerizing, since a structure can be reduced to a statically determinate form in more than one way.

**2. SLOPE DISPLACEMENT METHOD:**

It is displacement or equilibrium or stiffness method. It consists of series of simultaneous equations, each expressing the relation between the moments acting at the ends of the members is written in terns of slope & deflection. The solution of slope deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. Knowing these rotations, the end moments are calculated using slope deflection equations.

· **Limitations:**

1. This method is advantageous only for the structures with small Kinematic indeterminacy.

2. The solution of simultaneous equation makes the method tedious for annual computations.

The formulation of equilibrium conditions tends to be a major constraint in adopting this method.

*Hence flexibility coefficients & slope displacement methods have limited applications in the analysis of frames. While other methods like iterative or approximate methods are used for analyzing frames containing larger indeterminacy.*

**3. APPROXIMATE METHODS:**

Approximate analysis of hyper static structures provides a simple means of obtaining quick solutions for preliminary designs. It is a very useful process that helps to develop a suitable configuration for final (rigorous) analysis of a structure, compare alternative designs & provide a quick check on the adequacy of structural designs. These methods make use of simplifying assumptions regarding structural behavior so as to obtain a rapid solution to complex structures. However, these techniques should be applied with caution & not relied upon for final designs, especially complex structures.

The usual process comprises reducing the given indeterminate configuration to a structural system by introducing adequate number of hinges. It is possible to check the deflected profile of a structure for the given loading & there by locate the points of inflection.

Since each point of inflection corresponds to the location of zero moment in the structure, the inflection points can be visualized as hinges for purpose of analysis. The solution of the structure is rendered simple once the inflection points are located. In multistoried frames, two loading cases arise namely horizontal & vertical loading.

The analysis is carried out separately for these two cases:

· **VERTICAL LOADS:**

The stress in the structure subjected to vertical loads depends upon the relative stiffness of the beam & columns. Approximate methods either assumes adequate number of hinges to render the structure determinate or adopt simplified moment distribution methods.

· **HORIZONTAL LOADS:**

The behavior of a structure subjected to horizontal forces depends on its height to width ratio. The deformation in low-rise structures, where the height is smaller than its width, is characterized predominantly by shear deformations. In high rise building, where height is several times greater than its lateral dimensions, is dominated by bending action. There are two methods to analyze the structures subjected to horizontal loading.

**3.1. PORTAL METHOD:**

Since shear deformations are dominant in low rise structures, the method makes simplifying assumptions regarding horizontal shear in columns. Each bay of a structure is treated as a portal frame, & horizontal force is distributed equally among them.

**The assumptions of the method can be listed as follows:**

1. The points of inflection are located at the mid-height of each column above the first floor. If the base of the column is fixed, the point of inflection is assumed at mid height of the ground floor columns as well; otherwise it is assumed at the hinged column base.

2. Points of inflection occur at mid span of beams.

3. Total horizontal shear at any floor is distributed among the columns of that floor such that the exterior columns carry half the force carried by the inner columns.

**3.2. CANTILEVER METHOD:**

This method is applicable to high rise structures. This is based on the simplifying assumptions regarding the Axial Force in columns.

1. The basic assumption of the method can be stated as “the axial force in the column at any floor is linearly proportional to its distance from the centroid of all the columns at that level.

Assumptions 1&2 of the portal are also applicable to the cantilever method.

**3 POINTS OF INFLECTION METHOD:**

The frame is reduced to a statically determinate form by introducing adequate number of points of inflection. The loading on the frames usually comprises uniformly distributed dead loads & live loads.

**The following are assumptions made:-**

1. The beams of each floor act as continuous beams, with the points of inflection at a distance of one-tenth of the span from the joints.

2. The unbalanced beam moment at each joint is distributed equally among the columns at the joint.

3. Axial forces & deformations in beams are negligible.

**3.4 SUBSTITUTE FRAME METHOD:**

The method assumes that the moments in the beams of any floor are influenced by loading on that floor alone. The influence of loading on the lower or upper floors is ignored altogether. The process involves the division of multi-storied structure into smaller frames. These sub frames are known as equivalent frames or substitute frames.

The sub frames are usually analyzed by the moment distribution method, using only one cycle of distribution. The substitute frames are formed by the beams at the floor level under consideration, together with the columns above & below with their far ends fixed. The distributed B.M are not carried over far ends of the columns in this process; the moments in the columns are computed at each floor level independently & retained at that floor irrespective of further analysis.

**4. ITERATIVE METHOD:**

Iterative procedures form a powerful class of methods for analysis of indeterminate structures. These methods after elegant & simple procedure of analysis, that are adequate for usual structures.

These methods are based on the distribution of joint moments among members connected to a joint. The accuracy of the solution depends upon the number of iterations performed; usually three or five iterations are adequate for most of the structures.

The moment distribution methods were developed by Hardy Cross in 1930’s & by Gasper Kani in 1940’s. These methods involve distributing the known fixed moments of the structural members to the adjacent members at the joints, in order to satisfy the conditions of the continuity of slopes & displacements.

Though these methods are iterative in nature, they converge in a few iterations to give correct solution.

**4.1 MOMENT DISTRIBUTION METHOD:**

This method was first introduced by Prof. Hardy Cross is widely used for the analysis of intermediate structures. In this method first the structural system is reduced to its kinematically determinate form, this is accomplished by assuming all the joints to be fully restrained. The fixed end moments are calculated for this condition of structure. The joints are allowed to deflect rotate one after the other by releasing them successively. The unbalanced moment at the joint shared by the members connected at the joint when it is released.

**LIMITATIONS:**

1. This method is eminently suited to analyze continuous beams including non-prismatic members but it presents some difficulties when applied to rigid frames, especially when frames are subjected to side sway.

2. Unsymmetrical frames have to be analyzed more than once to obtain FM (fixed moments) in the structures.

3. This method can not be applied to structures with intermediate hinges.

**4.2 KANI’S METHOD:**

This method was introduced by Gasper Kani in 1940’s. It involves distributing the unknown fixed end moments of structural members to adjacent joints, in order to satisfy the conditions of continuity of slopes and displacements.

**ADVANTAGES:**

1. Hardy Cross method distributed only the unbalanced moments at joints, whereas Kani’s method distributes the total joint moment at any stage of iteration.

2. The more significant feature of Kani’s method is that the process is self corrective. Any error at any stage of iteration is corrected in subsequent steps.

Framed structures are rarely symmetric and subjected to side sway, hence Kani’s method is best and much simpler than pther methods like moment distribution method and slope displacement method.

**PROCEDURE:**

1. Rotation stiffness at each end of all members of a structure is determined depending upon the end conditions.

a. Both ends fixed

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

b. Near end fixed, far end simply supported

K_{ij}= ¾ EI/L; K_{ji}= 0

2. 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 including transitional moments, moment releases and carry over moments are computed for members and entered. The sum of the FEM at a joint is entered in the central square drawn at the joint).

3. Iterations can be commenced at any joint however the iterations commence from the left end of the structure generally given by the equation

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

4. Initially the rotational components ? Mji (sum of the rotational moments at the far ends of the joint) can be assumed to be zero. Further iterations take into account the rotational moments of the previous joints.

5. Rotational moments are computed at each joint successively till all the joints are processed. This process completes one cycle of iteration.

6. Steps 4 and 5 are repeated till the difference in the values of rotation moments from successive cycles is neglected.

7. Final moments in the members at each joint are computed 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 consideration by including column shear in the iterative procedure.

8. Displacement factors are calculated for each storey given by

Uij = -1.5 (Kij/?Kij)

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