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CE-416
Pre-stressed Concrete Lab Sessional
Presented By:
ABU SYED MD. TARIN
Id No:10.01.03.020
Course teachers:
Munshi Galib Muktadir
Sabreena Nasrin
Department Of Civil Engineering
Ahsanullah University of Science &
technology
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Stiffness method is an efficient way to solve complex
determinant or indeterminant structures . It will
introduced that is a modern method for structural
analysis. which is a powerful engineering method and
has been applied in numerous engineering fields such
as solid mechanics and fluid mechanics. it is also
called the displacement method.
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A statically indeterminate structure that the reaction
and internal forces cannot be analyzed by the application of
the equation of static alone . The indeterminacy of the
structure may be either external ,internal or both . The space
structute is externally indeterminate if the number of the
reaction components is more than six.
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Identify degree of kinematic indeterminacy (doki)
Apply restraints and make it kinematically determinate
Apply loads on the fully restraint structure and calculate
forces.
Apply unknown displacements to the structure one at a
time keeping all other displacements zero and calculate
forces corresponding to each dof.
Write the equilibrium equations. solve the equation in
matrix form and obtain the value of unknown
displacements.
Calculate other reactions.(by super position)
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The number of possible directions that displacements or forces
at a node can exist in is termed a degree of freedom (dof ). Some
examples are:
Plane truss: has 2 degrees of freedom at each node:
translation/forces in the x and y directions.
Beams: have 2 degrees of freedom per node: vertical
displacement/forces and rotation/moment.
Plane Frame: has 3 degrees of freedom at each node: the
translations/forces similar to a plane truss and in addition, the
rotation or moment at the joint.
Space Truss: a truss in three dimensions has 3 degrees of
freedom: translation or forces along each axis in space.
Space Frame: has 6 degrees of freedom at each node:
translation/forces along each axis, and rotation/moments about
each axis.
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For frame ,doki=8*3-(3+2)=19
For beam, doki=4*2-(2+1)=5
For truss, doki=6*2-(2+2+1)=7
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For the case of symmetry & antysymmetry, use of modified
stiffness makes the problem easier. Previously, it was
derived that stiffness factor =3EI/L when the far end is
hinged. This is also modified stiffness.
Modified Stiffness K’=2EI/L=K/2 (For symmetry).
K’=6EI/l=3k/2 (For Antisymmetry).
K’= 3EI/L (When Far End Hinged)
Determination Of Stiffness Factor
Case(i)When the far end is hinged: K=4EI/L.
Case(ii)When the far end is fixed : MAB=3EI/L.
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Direct Stiffness Method for Truss Analysis
The members are straight, slender, and prismatic. The crosssectional dimensions are small in comparison to the member
lengths.
The joints are assumed to be frictionless pins (or internal
hinges).
The loads are applied only at the joints in the form of
concentrated forces.
Direct Stiffness Method for Frame Analysis
The members are slender and prismatic. They can be straight or
curved, vertical, horizontal, or inclined.
The joints can be assumed to be rigid connection, frictionless
pins (or internal hinges), or typical connections.
The loads can be concentrated forces or moments that act at
joints or on the frame members, or distributed forces acting on
the members.
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Solution
In the first step identify the degrees of freedom of the
frame .The given frame has three degrees of freedom
(i) Two rotations as indicated by U1 and U2
(ii) One horizontal displacement of joint B and C as
indicated by U3
In the next step make all the displacements equal to
zero by fixing joints B and C as shown in Fig.23.5c. On
this kinematically determinate structure apply all the
external loads and calculate reactions corresponding
to unknown joint displacements .
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