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CE2302
STRUCTURAL ANALYSIS – CLASSICAL METHODS
3 1 0 4
OBJECTIVE
The members of a structure are subjected to internal forces like axial forces, shearing forces, bending and torsional
moments while transferring the loads acting on it. Structural analysis deals with analysing these internal forces in the
members of the structures. At the end of this course students will be conversant with classical method of analysis.
UNIT I
DEFLECTION OF DETERMINATE STRUCTURES
9
Principles of virtual work for deflections – Deflections of pin-jointed plane frames and rigid plane frames – Willot
diagram - Mohr’s correction
UNIT II
MOVING LOADS AND INFLUENCE LINES
(DETERMINATE & INDETERMINATE STRUCTURES)
9
Influence lines for reactions in statically determinate structures – influence lines for members forces in pin-jointed
frames – Influence lines for shear force and bending moment in beam sections – Calculation of critical stress resultants
due to concentrated and distributed moving loads.
Muller Breslau’s principle – Influence lines for continuous beams and single storey rigid frames – Indirect model
analysis for influence lines of indeterminate structures – Beggs deformeter
UNIT III
ARCHES
9
Arches as structural forms – Examples of arch structures – Types of arches – Analysis of three hinged, two hinged and
fixed arches, parabolic and circular arches – Settlement and temperature effects.
UNIT IV
SLOPE DEFLECTION METHOD
9
Continuous beams and rigid frames (with and without sway) – Symmetry and antisymmetry – Simplification for hinged
end – Support displacements.
UNIT V
MOMENT DISTRIBUTION METHOD
9
Distribution and carry over of moments – Stiffness and carry over factors – Analysis of continuous beams – Plane rigid
frames with and without sway – Naylor’s simplification.
TUTORIAL
15
TOTAL : 60
TEXT BOOKS
1.
“Comprehensive Structural Analysis – Vol. 1 & Vol. 2”, Vaidyanadhan, R and Perumal, P, Laxmi Publications,
New Delhi, 2003
2.
“Structural Analysis”, L.S. Negi & R.S. Jangid, Tata McGraw-Hill Publications, New Delhi, Sixth Edition, 2003
3.
Punmia B.C., Theory of Structures (SMTS ) Vol II laxmi Publishing Pvt ltd, New Delhi, 2004
REFERENCES
1.
Analysis of Indeterminate Structures – C.K. Wang, Tata McGraw-Hill, 1992

V SEMESTER CIVIL ENGINEERING
CE2302 STRUCTURAL ANALYSIS
NOTES OF LESSON
I UNIT – DEFLECTION OF DETERMINATE STRUCTURES
Theorem of minimum Potential Energy
Potential energy is the capacity to do work due to the position of body. A body of weight ‘W’ held at a height ‘h’
possess energy ‘Wh’. Theorem of minimum potential energy states that “ Of all the displacements which satisfy the
boundary conditions of a structural system, those corresponding to stable equilibrium configuration make the
total potential energy a relative minimum”. This theorem can be used to determine the critical forces causing
instability of the structure.
Law of Conservation of Energy
From physics this law is stated as “Energy is neither created nor destroyed”. For the purpose of structural analysis, the
law can be stated as “ If a structure and external loads acting on it are isolated, such that it neither receive nor
give out energy, then the total energy of the system remain constant”. With reference to figure 2, internal energy is
expressed as in equation (9). External work done We = -0.5 P dL. From law of conservation of energy Ui+We =0. From
this it is clear that internal energy is equal to external work done.
Principle of Virtual Work:
Virtual work is the imaginary work done by the true forces moving through imaginary displacements or vice versa. Real
work is due to true forces moving through true displacements. According to principle of virtual work “ The total virtual
work done by a system of forces during a virtual displacement is zero”.
Theorem of principle of virtual work can be stated as “If a body is in equilibrium under a Virtual force system and
remains in equilibrium while it is subjected to a small deformation, the virtual work done by the external forces
is equal to the virtual work done by the internal stresses due to these forces”. Use of this theorem for computation
of displacement is explained by considering a simply supported bea AB, of span L, subjected to concentrated load P at
C, as shown in Fig.6a. To compute deflection at D, a virtual load P’ is applied at D after removing P at C. Work done is
zero a s the load is virtual. The load P is then applied at C, causing deflection C at C and D at D, as shown in Fig. 6b.
External work done We by virtual load P’ is
. If the virtual load P’ produces bending moment M’, then the
internal strain energy stored by M’ acting on the real deformation d in element dx over the beam equation (14)
P' δ D
We
2
L M' M dx
M' dθ
; Ui
dUi
2
2 EI
0
0
0
U
L

Where, M= bending moment due to real load P. From principle of conservation of energy We=Wi
P' δD L M' M dx
2
2 EI
0
P
C
A
D
a
B
Fig.6a
x
P
P’
D
C
C
A
L
D
B
a
Fig.6b
x
L
If P’=1 then
M' M dx
EI
0
L
δD
(16)
Similarly for deflection in axial loaded trusses it can be shown that
n
δ
0
P' P dx
(17)
AE
Where,
= Deflection in the direction of unit load
P’ = Force in the ith member of truss due to unit load
P = Force in the ith member of truss due to real external load
n = Number of truss members
L = length of ith truss members.
Use
of
virtual
load
P’
Unit Load Method
Castiglione’s Theorems:
=
1
in
virtual
work
theorem
for
computing
displacement
is
called

Castigliano published two theorems in 1879 to determine deflections in structures and redundant in statically
indeterminate structures. These theorems are stated as:
1st Theorem: “If a linearly elastic structure is subjected to a set of loads, the partial derivatives of total
strain energy with respect to the deflection at any point is equal to the load applied at that point”
U
Pj j 1,2, ..... N (18)
δ j
2nd Theorem: “If a linearly elastic structure is subjected to a set of loads, the partial derivatives of total
strain energy with respect to a load applied at any point is equal to the deflection at that point”
U
δ j j 1,2,....... N (19)
Pj
The first theorem is useful in determining the forces at certain chosen coordinates. The conditions of equilibrium of
these chosen forces may then be used for the analysis of statically determinate or indeterminate structures. Second
theorem is useful in computing the displacements in statically determinate or indeterminate structures.
Betti’s Law:
It states that If a structure is acted upon by two force systems I and II, in equilibrium separately, the external
virtual work done by a system of forces II during the deformations caused by another system of forces I is equal
to external work done by I system during the deformations caused by the II system
I
II
Fig. 7
A body subjected to two system of forces is shown in Fig 7. W ij represents work done by ith system of force on
displacements caused by jth system at the same point. Betti’s law can be expressed as Wij = Wji, where Wji represents
the work done by jth system on displacement caused by ith system at the same point.

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