CE 302 UNIT 1
CE 302 UNIT 1
STRUCTURES 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 are
connected by rigid joints
Rigid joints: These are the joints which are capable of
transferring axial forces as well as moment. For example Joints
provided between RC beam and column
A rigid frame is completely analysed when the variation of AXIAL
FORCE, SHEAR and MOMENT along the lengths of all the members
are found
RIGID FRAMES PIN JOINTED FRAMES (TRUSSES)BEAMS
3. PIN JOINTED FRAMES (TRUSSES) – is a structure wherein all
members are considered to be connected by hinges, thus
eliminating shear and moments in the members,
A pin jointed frame or truss is completely analysed when the
axial forces in all the members are determined.
A Pin Joint is capable of transferring axial forces but can not
transfer moment. For example links of chain in cycle.
If body have no joints it is hard to break because there is no
plane of failure.
Basic Difference between Pin and Rigid joints:
In pin joints there is relative rotations between two members
whereas 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 statical
unknowns 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 redundants
should 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.
Methods of Structural Analysis
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
as
Exact method
Semi approximate Method
Approximate 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
Methods of Structural Analysis
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 structure
Examples: 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 Assumptions
Two Fundamental assumptions prevail
LINEAR 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, Δ
I
= Displ. due to PI alone
Δ
II
= Displ. due to PII alone
Δ
I+II
= total displacement due to combined action of PI & PII
applied in sequence PI and PII
Δ
II+I
= total displacement due to combined action of PI & PII
applied in sequence PIIand PI
Fundamental Assumptions
Two 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/Wang
Determine the variation of axial force. shear. and moment in all
members of the rigid frame shown in Fig. 2.3.2a.
Example 2.3.1/Wang
Determine 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 Deformation
1. (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 Theorm
Theorem I
Theorem II
3. The Moment Area Method
4. The Conjugate Beam Method
ANALYSIS OF INDETERMINATE STRUCTURES
METHOD OF ANALYSIS OF INDETRMINATE STRUCTURES
(Force Method of Analysis)
1. Method of Consistent Deformation
Force 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.
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 (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.
1. Method of Consistent Deformation Contd…
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.
Analysis of Statically Indeterminate Beams by the Force Method
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.
Analysis of Statically Indeterminate Beams by the Force
Method
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.
