The document discusses structural analysis concepts including: 1. Instability caused by single degree mechanisms, indeterminacy, and assumptions like small displacements and plane sections remaining plane. 2. Kinematic approaches are described for trusses, beams, and frames to determine degrees of indeterminacy. 3. Virtual work principles are used for displacement-based and force-based methods of structural analysis.

1. The ‘on-line’ content is being presented in four parts.
PART - I
1. Instability caused by Single Degree Mechanism
2. Indeterminacy
3. Assumptions
4. Trusses, Beams, and Hybrid systems
5. Virtual Work Principles

2. On-Line Material for Students,
Instructors, and Both
DYNAMIC ANALYSIS OF
SKELETAL STRUCTURES
Force and Displacement Methods,
Iterative Techniques

3. Why another Book on Structures?
• Kinematic Approach for structural stability and
Indeterminacy
• Basic Mechanisms and Independent Equations of
Equilibrium. Unified Energy method.
• Types of Displacements:
1. Small displacements for geometric linearity,
2. Negligible displacements compared to flexural,
Inextensibility. Unknowns of negligible magnitude and
other displacements –computational errors while
solving simultaneous equations.
3. Generalized Plastic Analysis. (mechanisms assume
small displacements).

4. The Foundation of the entire
analysis is based on The Two Basic
Assumptions

5. Small Displacements
validates superposition & geometric linearity
Plane section remains plane
valid for Shallow Beams, thin Plates, and Shells

6. Gifts of Plain section remains Plane
• Generalized stress-strain relation: M = EI (1/R)
• Flexural Formula: M/I = f/y = E/R, and
• The great Moment Area Theorems, for angle
Change and tangential deviation.

7. Non Linearity
Material Properties
• Elastic
Material Nonlinearity
• Elastic – Plastic
Geometric Nonlinearity
• P-Δ Effect
Δ P

8. Skeletal Members
Length, L >> Cross sectional dimensions represented by a, b
It is a versatile idealization. A consultancy of
the Domical Folded plate was constructed at
Wellington . The analysis was based on
Skeletal Idealization

9. VIRTUAL WORK PRINCIPLES
Displacement and Force Method
THE PRINCIPLES ARE EXTENSIVELY USED FOR:
1. Satisfying Equilibrium, and
2. Computing deformations, particularly the angle
discontinuities at the locations of the plastic
hinges (an idealized plastic analysis).
One may use the displacement diagram for obtaining the
corresponding equivalent load. From virtual work principle,
pj = At Km A. Displacement or Equilibrium Method.
The procedure retains symmetry of the stiffness matrix.
Similarly, from the Complementary Virtual Work:
dJ = [ B0 B]T Fm Pm ]. Force Method or Compatibility Method
U

10. Determinacy and Indeterminacy
• A determinate structure has the minimum constraints
• The number of excess constraints gives Degree of Indeterminacy, n.
Indeterminate Structure
• Equilibrium considerations, say, m= 2 j – 3 for plane hinged trusses
(provided it is stable). Illustrate from statics and kinematically.
• Rank Technique, r of coefficient matrix, and r’ of Augmented matrix (An
entire Article 4.10 is devoted to arrive at the independent equations of
equilibrium) . Releases and the corresponding redundants are chosen in
the process.
• Kinematic Approach: The next slides explain the procedure.

11. Kinematic Approach
Anatomy – Internal Forces
• Truss i=1
• Beams i=2 , (axial effect delinked)
• Frames i=3
• Grids i=2, (Girders of zero torsional stiffness)
i=3, (Torsionally stiff girders)
• Space frame i=6

12. TRUSS
• Assemble the Truss by adding the basic
triangles and reach the end span:
1. The number of members not included, their count give degree of
indeterminacy.
2. If the end of the span is not reached, the truss is a mechanism
(unstable). You should be able to sketch the mechanism. All joints hinged
Requirement: No stressing of the members

13. Additional Insight into the Mechanisms
(For Instructors)
Behavior
• With A as center of rotation of rigid body AEF, the nodes F
and E move in the directions as shown (clockwise)
• Similarly with B as center of rotation of rigid body DBG, it will
rotate clockwise, and the nodes D and G move in the
directions as shown without stressing any member. It is,
therefore, a “Mechanism”
• The Mechanism is identical , even if B is a roller support.
• The instantaneous center of rotation with B as roller support
is not different from the center of rotation when support B is
a hinge.
Consider hinge support at C instead of it being at B.
• The nodes E, D, G, and F can move in the directions as
shown. The rigid body DBCG when rotated clockwise or anti
clockwise, the corresponding movements are not possible
without stretching the members ED and FG. It is, therefore,
not a “MECHANISM”.
(The instructors may like to attempt the likely displaced
configurations, and thus explain the above concepts based on the
assumption of ‘small displacement ‘.
Trusses
A E D B
F G C
F G C
E D B

14. BEAMS
• Kinematic considerations applied to Beams
C
V
C
V
From Kinematic considerations: With the addition of two interior supports, n=2
With the introduction of the two hinge releases, it is restored as a determinate structure.
C
Geometric Loadings: If the support C settles by Δ, sketch the Deflected Shapes

15. On–Line material for Students
Loading sequence and Applied Tensegrity
Assignment
• ABCD is a beam. Other members are links
• Find degree of indeterminacy
• Analyze the beam-truss by Force method due
to elongation δ of the members BB’ and CC’ for
the properties as shown
• Introduce additional members, BC’ and B’C
with EA
• Analyze the truss for an applied udl equal to q
applied on the top beam ABCD
• The members AB’, B’C’, and C’D are high tensile
wires
• Assume appropriate value of E and strengths
for the beams, the struts and the cables.
Comment on the behavior of the structure
under the geometric and applied loading; and
its possible applications in a real life situation
Geometry
• A a B a C a D
I I I
½ A A, a A, a ½ A
B’ C’
A, a
The prismatic beam ABCD has the properties: EI, 2EA
Additional diagonal members, BC’ and B’C

16. Self Stressing
The concept of self stressing has nicely been used in
the construction of Cable domes for large spans
• CABLE DOMES CONSIST OF RING BEAMS, SELF STRESSED CABLES SEPARATED BY VERTICAL
STRUTS, AND A FABRIC CLADDING AS TENSION MEMBRANE
• WIND LOADING IS CRITICAL FOR THE FABRIC CLADDING AS
TENSION MEMBRANES
• ELIMINATES COSTLY ANCHORAGES REQUIRED FOR TENT LIKE MEMBRANES
• THE RING BEAMS BEING IN COMPRESSION, THESE MAY BE CAST IN REINFORCED CONCRRETE
• TOP RING BEAM USED SKY LIGHT IS IN TENSION
• TENSEGRITY WHILE ELIMINATING RING BEAMS, USE A LARGE NUMBER OF INCLINED STRUTS.

17. Typical Intermediate
ring beam
Bottom
Ring Beam
Top Ring Beam
Fabric as cladding