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AKTU,Lucknow
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INTRODUCTION TO FORCE
STRUCTURAL ANALYSIS
AND
DISPLACEMENT
METHODS
OF
Since twentieth century, indeterminate structures are being widely used for its obvious
merits. It may be recalled that, in the case of indeterminate structures either the reactions
or the internal forces cannot be determined from equations of statics alone. In such
structures, the number of reactions or the number of internal forces exceeds the number
of static equilibrium equations. In addition to equilibrium equations, compatibility equations
are used to evaluate the unknown reactions and internal forces in statically indeterminate
structure. In the analysis of indeterminate structure it is necessary to satisfy the
equilibrium equations (implying that the structure is in equilibrium) compatibility equations
(requirement if for assuring the continuity of the structure without any breaks) and
force displacement equations (the way in which displacement are related to forces). We
have two distinct method of analysis for statically indeterminate structure depending upon
how the above equations are satisfied:
1. Force method of analysis
2. Displacement method of analysis
In the force method of analysis,primary unknown are forces.In this method compatibility
equations are written for displacement and rotations (which are calculated by force
displacement equations). Solving these equations, redundant forces are calculated. Once the
redundant forces are calculated, the remaining reactions are evaluated by equations of
equilibrium.
In the displacement method of analysis,the
primary
unknowns
are
the
displacements. In this method, first force -displacement relations are computed and
subsequently equations are written satisfying the equilibrium conditions of the structure.
After determining the unknown displacements, the other forces are calculated satisfying
the compatibility conditions and force displacement relations The displacement-based
method is amenable to computer programming and hence the method is being
widely used in the modern day structural analysis.
DIFFERENCE BETWEEN FORCE & DISPLACEMENT METHODS
FORCE METHODS
1.
2.
3.
4.
DISPLACEMENT METHODS
1.
2.
3.
4.
Method of consistent deformation
Theorem of least work
Column analogy method
Flexibility matrix method
Types of indeterminacy- static indeterminacy
Slope deflection method
Moment distribution method
Kani’s method
Stiffness matrix method
Types
of
indeterminacy
indeterminacy-
kinematic
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Governing equations-compatibility equations
Force displacement
matrix
relations-
Governing equations-equilibrium equations
flexibility Force displacement relations- stiffness matrix
All displacement methods follow the above general procedure. The Slope-deflection and
moment distribution methods were extensively used for many years before the computer era.
In the displacement method of analysis, primary unknowns are joint displacements which are
commonly referred to as the degrees of freedom of the structure. It is necessary to consider
all the independent degrees of freedom while writing the equilibrium equations.These degrees
of freedom are specified at supports, joints and at the free ends.
SLOPE DEFLECTION METHOD
In the slope-deflection method, the relationship is established between moments at the ends
of the members and the corresponding rotations and displacements.
The slope-deflection method can be used to analyze statically determinate and indeterminate
beams and frames. In this method it is assumed that all deformations are due to bending only.
In other words deformations due to axial forces are neglected. In the force method of analysis
compatibility equations are written in terms of unknown reactions. It must be noted that all
the unknown reactions appear in each of the compatibility equations making it difficult to
solve resulting equations. The slope-deflection equations are not that lengthy in comparison.
The basic idea of the slope deflection method is to write the equilibrium equations for each
node in terms of the deflections and rotations. Solve for the generalized displacements. Using
moment-displacement relations, moments are then known. The structure is thus reduced to a
determinate structure. The slope-deflection method was originally developed by Heinrich
Manderla and Otto Mohr for computing secondary stresses in trusses. The method as used
today was presented by G.A.Maney in 1915 for analyzing rigid jointed structures.
Fundamental Slope-Deflection Equations:
The slope deflection method is so named as it relates the unknown slopes and deflections to
the applied load on a structure. In order to develop general form of slope deflection
equations, we will consider the typical span AB of a continuous beam which is subjected to
arbitrary loading and has a constant EI. We wish to relate the beams internal end moments
in terms of its three degrees of freedom, namely its angular displacements
and linear displacement which could be caused by relative settlements between
the supports. Since we will be developing a formula, moments and angular displacements
will be considered positive, when they act clockwise on the span. The linear displacement
will be considered positive since this displacement causes the chord of the span and the
span’s chord angle to rotate clockwise. The slope deflection equations can be obtained by
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using principle of superposition by considering separately the moments developed at each
supports due to each of the displacements
&
Case A: fixed-end moments
,
=
,
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,
Case B: rotation at A,
(angular displacement at A)
Consider node A of the member as shown in figure to rotate while its far end B is fixed. To
determine the moment
needed to cause the displacement, we will use conjugate beam
method. The end shear at A` acts downwards on the beam since
is clockwise.
,
Case C: rotation at B,
(angular displacement at B)
In a similar manner if the end B of the beam rotates to its final position,
while end A is
held fixed. We can relate the applied moment
to the angular displacement
and the
reaction moment
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