ce 302 unit 1

CE 302 UNIT 1

CE 302 UNIT 1STRUCTURES Most structures fall under the following categories

Examples: Bridges, Dams, Buildings, Walls, Towers, Shells, Cables,Folded Plates, Silos, Slopes and Retaining Walls etc

1. BEAMS – Structural member subjected to transverse loads only.Analysis is complete when the BMD and SFD is – known

2. RIGID FRAMES – Structure composed of members which areconnected by rigid jointsRigid joints: These are the joints which are capable oftransferring axial forces as well as moment. For example Jointsprovided between RC beam and column

A rigid frame is completely analysed when the variation of AXIALFORCE, SHEAR and MOMENT along the lengths of all the membersare found

RIGID FRAMES PIN JOINTED FRAMES (TRUSSES)BEAMS

3. PIN JOINTED FRAMES (TRUSSES) – is a structure wherein allmembers are considered to be connected by hinges, thuseliminating shear and moments in the members,

A pin jointed frame or truss is completely analysed when theaxial forces in all the members are determined.

A Pin Joint is capable of transferring axial forces but can nottransfer moment. For example links of chain in cycle.

If body have no joints it is hard to break because there is noplane of failure.

Basic Difference between Pin and Rigid joints:In pin joints there is relative rotations between two memberswhereas in rigid joints it is not possible.

Methods of Structural Analysis

Every method of structural analysis can be classified as either a force method or a displacement method.

Force method For statically determinate structures, the internal forces within the constituent members can be determined by the laws of statics alone at the very beginning, and then the deformed shape of the structure follows. For statically indeterminate structures, the relative sizes of the constituent members are required in the solution for the redundants(unknown forces/reactions in excess of equilibrium eqns) from the conditions of consistent deformation. The remaining staticalunknowns are obtained after the redundants are determined.Examples: Consistent Deformation method, Flexibility

Method, column-analogy method, the three-moment equation

Methods of Structural Analysis Contd…..

Force method The force method of analysis can be derived entirely from the physical conditions of consistent deformation along the lines of action of the redundants, or it can be derived entirely from an elegant theorem which states that the redundantsshould be such as to keep at a minimum the total strain energy within the structure.

Displacement method In the displacement method, whether the structure is statically determinate or statically indeterminate, the solution procedure is the same; that is, the displacements of the joints in the structure are solved at the very beginning from an equal number of equations of equilibrium. Only then are the internal forces within the constituent members and the external reactions acting on the whole structure determined from the deformed shape of the structure.

Examples: Moment distribution method, slope-deflection Method, Stiffens method

Displacement method -- Because of the easiness with which matrix operations can be performed on the electronic computer, the matrix displacement method has become the most modern and efficient method for very large structures, especially in the final-check analysis stage.


Based upon, whether in the analysis, the effects of axial forces, shear forces, bending moments and torsion have been considered or not, a method of analysis can be further classified asExact methodSemi approximate MethodApproximate Method

Exact method Matrix Methods of Analysis (Flexibility Methods & Stiffness Methods wherein effects of all the forces viz axial forces, shear forces, bending moments and torsion have been considered), are falling under this category


Semi approximate Method These methods do not take into account the simaltaneous effects of all these forces viz axial forces, shear forces, bending moments and torsion. Generally these methods consider the effects of BM only.Examples: Slope deflection method, Moment Distribution method and Kani’s menthod

Approximate Method These methods involve more aasumptions to simplfy the problem. These methods do not consider the behavior of overall structureExamples: Substitute frame method, portal method and cantilever method

In spite of the need for the formal and more exact method of analysis, approximate methods are desirable for purposes of estimation, at least in the search for relative sizes of members. Fundamental AssumptionsTwo Fundamental assumptions prevailLINEAR The word linear refers to the material property that the relationship between stress and strain is linear. The implication is that nowhere in the structure is the stress above the proportional limit. When the response of a structure is sought wherein the material is stressed beyond the proportional limit, the method of analysis falls into the realm of nonlinear analysis, which is certainly dependent on the shape of the stress-strain curve beyond the proportional limit. Nonlinear analysis is beyond the scope of this discussion

Fundamental Assumptions

Linearity ---means that the internal stresses and the resulting displacements increase in proportion to the external forces. In Fig. the external load P acting on the structure and the resulting displacement Δ at any point of the structure are plotted along the vertical and horizontal axes respectively. The structure is said to behave linearly if the load displacement relationship is represented by the straight line OA.

PRINCIPLE OF SUPERPOSITION According to this principle, the total response of a structure on account of the combined action of any two systems of external forces PI and PII is equal to the sum of the responses due to the two systems of forces acting separately. Thus, referring to Fig. 2.2,

Δ I+II = Δ II+I = Δ I + Δ II , Where, ΔI = Displ. due to PI aloneΔII = Displ. due to PII alone

Δ I+II = total displacement due to combined action of PI & PIIapplied in sequence PI and PII

Δ II+I = total displacement due to combined action of PI & PII

applied in sequence PIIand PI

Fundamental AssumptionsTwo fundamental assumptions prevail. NO AXIAL DEFORMATION The other fundamental assumption is that

the length of a straight segment within the structure is not affected by its curvature during deformation, nor is it affected by displacement of its ends in the transverse direction.

Example 2.3.1/WangDetermine the variation of axial force. shear. and moment in all members of the rigid frame shown in Fig. 2.3.2a.

Common Methods for Determination of Deformation1. (A) The Unit Load Method ----- Beam Defelction

Where M = BM due to external Loads Only &m= BM due to unit load at C in the direction of Δ

(B) The Unit Load Method ----- Beam Slopes

Where M = BM due to external Loads Only &m= BM due to unit Moment at C in the direction of θ

Common Methods for Determination of Deformation Contd…

2. The Partial Derivative Method ----- Castiglianos TheormTheorem ITheorem II

3. The Moment Area Method4. The Conjugate Beam Method

ANALYSIS OF INDETERMINATE STRUCTURES

METHOD OF ANALYSIS OF INDETRMINATE STRUCTURES(Force Method of Analysis)

1. Method of Consistent DeformationForce response of statically determinate beams, rigid frames, and trusses can be determined solely by the laws of statics, and in the solution procedure the properties of members, such as moments of inertia for beams and rigid frames or areas for trusses, are not required.

The deformation response can be obtained after the force response, but in the solution procedure, member properties are required.

1. Method of Consistent Deformation Contd…

When a structure-whether it be a beam, a rigid frame, or a truss-is statically indeterminate, the force response cannot be determined by the laws of statics alone. In these situations, some unknown reactions, or member forces, equal in number to the degree of indeterminacy, can be regarded as unknown forces (Redundants) acting on a basic determinate structure, and their magnitudes can be obtained at the very beginning from the conditions of consistent deformation (Condition of Geometry). In establishing these conditions of geometry, member properties are then required.


When a structure-whether it be a beam, a rigid frame, or a truss-is statically indeterminate, the force response cannot be determined by the laws of statics alone. In these situations, some unknown reactions, or member forces, equal in number to the degree of indeterminacy, can be regarded as unknown forces (called as redundant) acting on a basic determinate structure, and their magnitudes can be obtained at the very beginning from the conditions of consistent deformation.


Analysis of Statically Indeterminate Beams by the Force Method

For the coplanar parallel-force system acting on a beam, there are two independent conditions of statics, with one more for each internal hinge present in the beam. The number of excess reactions over that of independent equations of equilibrium is the degree of indeterminacy, or

NI = NR -2-NIH

in which NI is the degree of indeterminacy, NR is the total number of reactions, and NIH is the number of internal hinges in the beam.


Following steps are to be adopted in Method of Consistent Deformation.

STEP1: choose the redundant reactions,

STEP II: remove the physical restraints associated with the redundant reactions, and obtain a basic determinate beam subjected to the combined action of the applied loads and the unknown redundant reacting forces.

If a simple support is removed and the reaction is replaced by an unknown reacting force, the condition of geometry is that the deflection there must be zero. If a fixed support is changed into a simple support and the original moment reaction is replaced by an unknown reacting moment, the condition of geometry is that the slope there must be zero.


If a fixed support is completely removed and the original restraint is replaced by an unknown reacting force and an unknown reacting moment, the two conditions of geometry are that both the deflection and the siope there must be zero.

ce 302 unit 1

Documents