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CE-416Pre-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

Statically indeterminate structure: stiffness method

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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.

Introduction

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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.

Statically indeterminate structure

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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)

Analysis steps of stiffness method

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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.

Degree of Freedom

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Degree of kinematic indeterminacy

doki=(no of joints)*(no of dof)-no of restraint

For frame ,doki=8*3-(3+2)=19For beam, doki=4*2-(2+1)=5For 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.

Modified Stiffness

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Direct Stiffness Method for Truss AnalysisThe members are straight, slender, and prismatic. The cross-

sectional 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 AnalysisThe 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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ExampleAnalyze the plane frame shown in Fig.23.5a by the direct stiffness method. Assume that the flexural rigidity for all members is the same .Neglect axial displacements.

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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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