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- 1. 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 12/4/2013 1
- 2. 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. 12/4/2013 2
- 3. 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. 12/4/2013 3
- 4. 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) 12/4/2013 4
- 5. 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. 12/4/2013 5
- 6. 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 12/4/2013 6
- 7. 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. 12/4/2013 7
- 8. 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. 12/4/2013 8
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- 10. 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 . 12/4/2013 10
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