Matrix Structural Analysis - 1D Finite Elements

Course Description:

The Direct Stiffness Method in plane truss, plane frame, spatial truss, spatial frame. Rigid Οffsets. Internal Releases. Beams of variable cross section. 1D Finite Elements.

Prerequisite Knowledge

Students is recommended to have basic knowledge in Mechanics of the rigid body, Mechanics οf Deformable Solids, Structural Analysis of Determined Structures, Structural Analysis of Indetermined Structures

Course Units

# Title Description Hours
1 Introduction Overview of matrix structural analysis and design. Primary structural members and their modeling – Matrix structural analysis steps. Flexibility – stiffness methods. Computer programs and rational use. Basic steps of programming the direct stiffness method. Degrees of freedom of plane and spatial structures. 2
2 Plane truss Global and local systems of axes. Vectors of end-actions and end-translations of a plane truss element. Transformation matrix. Calculation of local-global stiffness matrix of a plane truss element: analytical and numerical (shape function, deformation matrix) methods. Vectors of nodal-forces and nodal-translations, global stiffness matrix of a plane truss. Modification of global stiffness matrix due to support conditions – Reordering matrix. Modification of global stiffness matrix of a plane truss due to inclined and elastic supports. Plane truss subjected to member loading. Restrained – equivalent structure. Stress resultants of plane truss members. 12
3 Plane frame Vectors of end-actions and end-displacements of a plane frame element. Transformation matrix. Calculation of local - global stiffness matrix of a plane frame element: analytical and numerical (shape functions, deformation matrix) methods. Vectors of nodal-forces and nodal-displacements, global stiffness matrix of a plane frame. Modification of global stiffness matrix due to support conditions – Reordering matrix. Modification of global stiffness matrix of a plane frame due to inclined and elastic supports. Plane frame subjected to member loading. Restrained – equivalent structure. Stress resultants of plane frame members. 10
4 Spatial truss Transformation matrix of a spatial truss element. Local - global stiffness matrices of a spatial truss element: analytical and numerical methods. Steps of analysis of a spatial truss. 4
5 Spatial frame Transformation matrix of a spatial frame element. Basic transformation matrix. Transformation matrix with special orientation. Transformation matrix for special auxiliary point. Formulation of transformation matrices of elements of other type of skeletal structures. Local stiffness matrix of a spatial frame element. Formulation, stiffness terms. Formulation of local stiffness matrices of members of all other types of skeletal structures. Vectors of nodal-actions and nodal-displacement of a spatial frame. 4
6 Grid Analysis of a grid structure. Solving a grid structure as a special case of a spatial framed structure. 4
7 Rigid joints Kinematic relations and equivalent actions between two points of a rigid body plane. Rigid joints in plane framed structure. Kinematic relations and equivalent actions between two points of space rigid structure. Rigid joints in space frame element. 8
8 Internal releases Combined node method. Degrees of freedom of combined nodes. Assembly of total global stiffness matrix with combined nodes. Computation of nodal actions of restrained and equivalent structures with combined nodes. Elastic hinge. Internal releases– Method of modified stiffness matrices. Modified matrices and internal releases. Restrained actions – Equivalent actions. Static condensation method. Physical interpretation of static condensation. Qualitative examination of the stiffness coefficients of a hyper-element. Stiffness matrix and restrained actions with elastic hinge 8
9 Elements of variable cross-section Stiffness matrix– Analytic evaluation and approximate computation. Restrained actions. Analytic evaluation and approximate computation. 4

Learning Objectives

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

1. know through matrix view, the method of nodal displacement,
2. they are aware of the analysis of structures and of the developing intensive actions,
3. understanding the necessary theoretical background for writing a software code to solve skeletal structures.

Teaching Methods

 Teaching methods Lectures in class. Solve examples and problems in the classroom. Presentations in the Table. Slides Power Point. Demonstrate an application to a 1D framed structure with the help of free software on the internet. Students solve the problems with the help of teachers using EXCEL and MATLAB on PC. Students are examined in the solution of two structures (plane truss, plane frame) employing the Direct Stiffness Method.

Student Assessment

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

Textbooks - Bibliography

1. Papadrakakis M. and Sapountzakis E.J. (2017) Matrix Methods for Advanced Structural Analysis, Butterworth – Heinemann, Elsevier, ISBN 978-0-12-811708-8.
2. Παπαδρακάκης Μ. Σαπουντζάκης Ε. (2015). Ανάλυση Ραβδωτών Φορέων με Μητρωϊκές Μεθόδους – Μέθοδος Άμεσης Στιβαρότητας, Τσότρας.
3. Κωμοδρόμος Π. (2009). Ανάλυση Κατασκευών, Παπασωτηρίου.

Lecture Time - Place:

• Tuesday, 12:45 – 14:30,
Rooms:
• Αιθ. 17
• Αιθ. 5
• Thursday, 12:45 – 14:30,
Rooms:
• Αιθ. 3
• Ζ. Κτ. 1 Πολ., Αιθ. 17

© 2017 School of Civil Engineering, ΕΜΠ