# Structural Analysis of Statically Indeterminate Structures

## Course Description:

Structural analysis of statically indeterminate structures. Forces Method and Nodal Displacement Method. Influence Lines for indeterminate structures (Qualitative)

• Semester 5
• Teaching hours 4
• Instructors

### Prerequisite Knowledge

Engineering Mechanics I (Rigid Body Statics), Engineering Mechanics II (Mechanics of Deformable Bodies), Statics I (Statically Determinate Structures).

### Course Units

# Title Description Hours
1 Force Method -Introduction Difference between statically determinate and indeterminate structures. Equilibrium of Forces and compatibility of deformations. Static and kinematic duality. Static Indeterminacy. Formulation of the force method in systems of one-degree indeterminacy. 4
2 Solution of single and two DOF system Applications to structures of one-degree indeterminacy selecting alternative redundants. 4
3 Calculation of deformations Formulation of the Force method for multi-degree indeterminate structures. Evaluation of deformations in the primal – statically determinate system due to external load and flexibility coefficients by applying the unit load theorem. Reciprocity of Deformations, Betti-Maxwell Principle. 4
4 Temperature changes Solving problems with two redundant forces. Criteria for selecting effective redundants. Solving problems accounting for temperature variations. 4
5 Support retreat - Elastic supports Support settlements. Solution of relevant problems. Elastic supports. Applications. Qualitative sketch of moment and force diagrams. 4
6 Symmetrical structural systems Symmetric structures about an axis. Symmetric and anti-symmetric loads. Symmetry Propositions for the whole or half of the structure. Solution of problems. 4
7 Selection of Statically Accepted Possible Status Calculating the deformations of indeterminate structures. Unit Load Theorem using a statically admissible distribution of moments and forces. Compatibility checks of the solutions. 4
8 Introduction to the Nodal Displacements Method Kinematic indeterminacy of structural systems. Nodal Displacements, degrees of freedom, DOFs. Neglecting axial deformations. Consideration of mixed fixation (fixed end and pinned –fixed beams). Formulation of the method of nodal displacements for single-DOF systems. 4
9 Dual consideration with Force Method Formulation of the Nodal Displacements method of in multi-DOF systems. Dual consideration using the Forces Method. 4
10 Fundamental solutions Fundamental solutions of fixed end beam and fixed – pinned beam, end displacements and rotations. Stiffness Coefficients. Examples. 4
11 Structural systems with oblique members Structural systems with oblique members and geometrically coupled displacements. Investigation of appropriate equilibrium equations for the displacements. Examples. Symmetric loads. 4
12 Temperature changes. Resignations of supports. Elastic supports Temperature variations. Support settlements. Elastic supports. Examples. 4
13 Qualitative scribing of M,Q,N forces diagrams & Influence lines Qualitative sketch of moment and force diagrams. Simple and more complex indeterminate structural systems. Influence Lines of indeterminate structural systems. Muller-Breslau principle. Applications - Qualitative sketch of influence lines for reactions, moments and shearing and axial forces. 4

### Learning Objectives

Upon successful completion of the course, students will be able to:

1. know the basic principles of static and kinematic indeterminancy,
2. realize the very basic concept of compatibility of deformations. Emphasis on the forces method is given in the equilibrium and the demand for the compatibility of deformations in the direction of the redundant quantities. Reversely, nodal displacements method relies on the compatibility of deformations and enforces equilibrium in the direction of the unknown displacements.
3. understand the behavior of relatively complex indeterminate structural systems,
4. formulate the computational model of relatively complex indeterminate structural systems, and
5. analyze manually stress and strain state of indeterminate structural systems, qualitative sketching of M,Q,N forces diagrams and the influence lines of indeterminate structural systems.

### Teaching Methods

 Teaching methods Class lectures and workshops. Theory, Applications and Examples worked in the class by two tutors in two parallel sections. Yes. One to two applications in small-scale systems related with the evaluation of deformations of frame structures for which students have already worked out the analytical part. Yes. Yes. Solving one or two indeterminate structural systems and determining M,Q,N force diagrams using the forces and/or nodal displacements method.

### Student Assessment

• Final written exam: 70%
• Assignments (projects, reports): 20%
• Problems - Applications: 10%

### Textbooks - Bibliography

1. Στατική των Κατασκευών Τομός ΙΙ – Υπερστατικοί Φορείς Κλασικές Μέθοδοι Ανάλυσης Αβραμίδης Ι.Ε. Εκδόσεις Σοφία Α.Ε. 2007 Θεσσαλονίκη.
2. Στατική των Δομικών Φορέων, Μέρος Α’, Λ.Θ. Σταυρίδης, Εκδόσεις Κλειδάριθμος 2008.
3. Στατική των Γραμμικών Φορέων Βαλιάσης Θ. Εκδόσεις Ζήτη 1997 Θεσσαλονίκη.
4. Εφαρμοσμένη Στατική W. Wagner G. Erlhof Εκδόσεις Κλειδάριθμος 2012 Αθήνα.
5. Σημειώσεις «Παραδείγματα υπολογισμού υπερστατικών φορέων με τη μέθοδο των παραμορφώσεων» Σπηλιόπουλος Κ. Εκδόσεις Ε.Μ.Π. 1997 Αθήνα.

## Lecture Time - Place:

• Monday, 09:45 – 12:30,
Rooms:
• Αιθ. 17
• Ζ. Κτ. 1 Πολ., Αιθ. 12
• Wednesday, 10:45 – 12:30,
Rooms:
• Αιθ. 17
• Ζ. Κτ. 1 Πολ., Αιθ. 12

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