The Finite Element Method

1
The Finite Element Method

• Dr. Abdul Razzaq Touqan
• Department of Civil Engineering

2
Introduction to the Finite Element Method

• Objectives
• Students will be introduced to FE

3
Finite Element Method Defined

• Problems are governed by differential or integral
  equations.
• These equations provide an exact, closed-form
  solution (analytical).
• Complexities in geometry, properties and in the
  boundary conditions prevent ability to obtain
  exact solutions. Also 2D and 3D elements are
  governed by partial differential equations, no
  closed form solution exists for such elements..

4
Finite Element Method Defined (cont.)

• Complex regions are discretized (anatomized)
  into simple shapes called elements. The
  continuum has infinite number of DOFs.
  Discretized model has finite number of DOFs.
• The elements are either 1D, 2D or 3D.
• Properties and relationships are analogized over
  the elements in terms of unknown values at nodes.
• A set of linear/nonlinear equations are obtained
  by linking the individual elements and
  considering effects of loads and boundary
  conditions.

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Finite Element Method Defined (cont.)

• Number of equations is large ? requires digital
  computers otherwise worthless.
• Advances in computers and software has brought
  the FEM within reach for engineers and students.

6
Simplified example of concept of FE finding the
perimeter of a circle

• 1. Analytical integral equation of infinitely
  small arcs
• 2. Analogical approximate arc into straight
  lines and then perimeter is between (4v2)r and
  8r! How?
• 3. Anatomical (Finite element) combine 1 and 2
  discretize circle into finite number of divisions
  and approximate curved arcs by straight lines
  p(2rsin?/2)n, ? 2p/n

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Origins of the Finite Element Method

• Basic concepts evolved over a period of 150 or
  more years.
• The term finite element was first coined by
  Clough in 1960.
• The first book on the FEM by Zienkiewicz and
  Chung was published in 1967.
• In the late 1960s and early 1970s, FEM was
  applied to a wide variety of engineering problems.

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Origins of the Finite Element Method (cont.)

• 1970s marked advances in mathematical treatments
• Most FEM software originated in the 1970s and
  1980s.
• FEM is one of the most important developments in
  computational methods in the 20th century.
• In few decades, the method has evolved to cover
  many scientific and technological areas (solid
  and fluid mechanics, chemical reactions,
  electromagnetics, biomechanics, heat transfer and
  acoustics ).

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Advantages of the Finite Element Method

• Handle complex geometry heart and power of it.
• Handle complex restraints indeterminate
  structures
• Handle complex loading point, pressure, inertia
• Handle nonhomogeneous material bodies every
  element assigned a different material properties.

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Advantages of the Finite Element Method (cont.)

• Handle nonisotropic materials bodies
  orthotropic, anisotropic
• Handle special material effects temperature
  dependent properties, plasticity, creep,
  swelling,
• Handle complex analysis types vibration,
  nonlinear..
• Model special geometric effects large
  displacements and large rotations

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Disadvantages of the Finite Element Method

• Handle specific problems, thus no general
  closed-form solution
• Approximate the mathematical model (the source of
  so-called inherited errors.)
• Need experience and judgment to construct a good
  finite element model.
• A computer and reliable FEM software are
  essential.
• Large amount of Input and output data to prepare
  and interpret.

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Disadvantages of the Finite Element Method (cont)

• Numerical problems
• Computers carry a finite number of significant
  digits? round off and error accumulation.
• Susceptible to user-introduced modeling errors
• Poor choice of element types.
• Distorted elements.
• Geometry not adequately modeled.
• Certain effects not automatically included
• Buckling, large deflections and rotations,
  material and geometric nonlinearities

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homework

• Pick a small problem in engineering. Then provide
  analytical, analogical and anatomical (FE)
  solutions for it. Pick it up from a library
  reference if not able to visualize one.

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End of introduction to FE
Let Learning Continue
15

• Finite Element Analysis of Framed Structures

16
General Structural Problem

• For the following structure, we want to compute
• Bending moments, shear forces, axial forces
• Bending, shear and axial stresses and strains
• Deflections

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Finite Element Analysis (FEA), The basic idea

• Complex structures idealized through mathematical
  models are too complicated to derive
  relationships between applied loads, deflections
  and internal stresses.
• Hence discretize (anatomize) into many individual
  finite elements of simpler form, e.g. a beams or
  columns
• Determine the relationship between load,
  displacement, stresses and strains within a
  finite element (analogize)
• Assemble elements to satisfy equilibrium and
  compatibility (analytical solution).

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Modelling Idealization
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Analysis of bar elements

• Analytical solution review mechanics of
  materials

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Analysis of bar elements (cont.) analytical
solution

• but
• for constant A
• Homogeneous sol. particular solution

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Analysis of bar elements (cont.)analytical
solution example

• -analyze the following structure

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Analysis of bar elements (cont.)analytical
solution example

• -solution

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Analysis of bar elements (cont.)analytical
solution example
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For a finite element, we need to derive the
relationship between

• External Loads
• Deflections/deformations
• Internal stresses and strains

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General basic steps for finite element
method applied here for bar elements

• Input
• Problem to be solved
• Physics of problem
• Mathematical model

• Processing
• Propose theory
• Formulate equations
• Solve equations

Output 1. Verify compatibility 2. Verify
equilibrium 3. Verify stress-strain relations
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Input

• Problem analysis of a bar subjected to axial
  loads
• Physics bar subjected to axial
  stress\deformation
• Model define material, geometry and loading

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Processing 1. propose theory for solution

• a. Select element type axial, 2-node
• b. Select a displacement function
• (1) u generic displacement

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2.1 propose theory for solution b. Select
element type and displacement functions

• b. con express u(x) in terms of nodal
  displacements using boundary conditions.
• (2)

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2.1 propose theory for solution b. Select
element type and displacement functions

• Sub (2) into (1)

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2.1 propose theory for solution

• c. Derive strain and stress displacement
  relationships

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2.2 formulate equations a. Derive the element
stiffness matrix and equations using principle of
Virtual Work (general derivation)

• (fj joint, fb equivalent joint)
  loads

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2.2 formulate equations a. Derive the element
stiffness matrix and equations
for uniform axial load b
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2.2 formulate equations

• b. Transfer equations from local to global axes
  (later)
• c. Assemble element equations to obtain global
  equations and introduce boundary conditions

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2.3 Solve equations provide details

• a. Solve for the unknown DOF
• b. Solve for reactions
• c. Solve for element strains and stresses

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3. Output

• Verify compatibility same as analytical
• Verify equilibrium reaction3X515KN ok
• Verify stress strain relationship
• -stress is constant unlike true solution which is
  linear

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Ways of trial improvements?

• Use more subdivisions
• Use more terms, three node element (homework)
• c. Replace terms of homogenous with higher
  order terms
• Check each idea from above, propose more ideas,
  carry out conclusions!

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End of FE for bars
Let Learning Continue
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• Ways of trial improvements?

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General basic steps for FE methodreminder

• Input
• Problem to be solved
• Physics of problem
• Mathematical model

• Processing
• Propose theory
• Formulate equations
• Solve equations

Output 1. Verify compatibility 2. Verify
equilibrium 3. Verify stress-strain relationships
42
Ways of trial improvements?

• Use more subdivisions
• Use more terms, three node element (homework)
• c. Replace terms of homogenous with other
  terms
• Check each idea from above, propose more ideas,
  carry out conclusions!

43
Ways of trial improvements?a.Use more
subdivisions

• To compare with previous solution
• Input, no change
• Processing
• Propose theory for solution
• A. Select element type no change, two elements
• B. Displacement function no change
• C. Stress-strain/displacement relations no
  change
• Formulate equations
• Element stiffness and equivalent nodal loads
  change L3m to L1.5m
• Local to global no change
• Assemble next

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2.2 formulate equations

• c. Assemble the element equations and introduce
  boundary conditions

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2.3 Solve equations

• a. Solve for the unknown DOF
• b. Solve for reactions

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2.3 Solve equations

• c. Solve for element strains and stresses

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3. Output

• Verify compatibility same as analytical
• Verify equilibrium reaction3X515KN ok
• Verify stress strain relationship more
  subdivision produces more accuracy.

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Ways of trial improvements? C. Replace some
terms with higher order terms

• To compare with original solution
• Input, no change
• Processing
• Propose theory for solution
• a. Discretize No change
• b. Element type and displacement functions

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Processing2.1 Propose theory

• c. Stress-strain/displacement relations

50
2.2 Formulate equations

• Derive element stiffness and equivalent joint
  loads

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2.2 formulate equations

• b. Transfer equations from local to global no
  change
• c. Assemble element equations and introduce
  boundary conditions

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2.3 Solve equations

• a. Solve for the unknown DOF
• b. Solve for reactions

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2.3 Solve equations

• c. Solve for element strains and stresses

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3. Output

• Verify compatibility different than analytical
• Verify equilibrium reaction3X515KN ok
• Verify stress strain relationship
• -stress is linear, however it is completely wrong
  in values and slope

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Sources of Error in the FEM (cont.)

• Three sources modeling (idealization
  formulation and system), discretization and
  solution (numerical) errors

• 1. Modeling errors result from
• -Either the use of elements or systems that
  don't precisely describe the behavior of the
  physical problem,
• -Elements which are used to model physical
  problems for which they are not suited are
  sometimes referred to as ill-conditioned or
  mathematically unsuitable elements.
• -Using 1D or 2D instead of 3D systems

57
Sources of Error in the FEM (cont.) 1. Modeling
error

• Example a FE with displacements varying in a
  linear manner will produce no element modeling
  error if used for linearly varying physical
  problem, but would create a significant error if
  used to represent a quadratic or cubic varying
  displacement field.

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Sources of Error in the FEM

• 2. Discretization error can be related to
  modeling the boundary shape, the boundary
  conditions, etc.

Error due to poor geometry representation.
error effectively eliminated
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Sources of Error in the FEM

• 3. Solution (numerical error) includes truncation
  errors and round off errors. Numerical error is
  therefore a problem mainly concerning the FEM
  buyers and developers.
• The user can also contribute to the numerical
  accuracy, for example, by specifying a physical
  quantity, say Youngs modulus, E, to an
  inadequate number of decimal places.

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Conclusions

• Discuss further ideas
• Continuous within and between elements
  (compatibility) Co enforce displacement
  continuity across common boundaries (example bar
  element). C1 enforce both displacement and its
  first derivative continuous across common
  boundaries (example beam element).
• Complete for equilibrium
• Rigid body motion for real
• Also discuss acceptance criterion (verification)

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SAP acceptance criterion (documentation\verificati
on\methodology)
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End of ways of improvements
Let Learning Continue
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• Theory of plane and space trusses

64
Local versus global coordinates

• Element axes are not all the same.
• So there is a need for a coordinate
  transformation

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Coordinate transformations from local to global
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Coordinate transformations from local to global
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Coordinate transformations from local to global

• Transformation matrix from global to local T is
• Notice for an orthogonal matrix, the inverse
  transformation from local to global is quite easy
  because

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Coordinate transformations from local to global

• -The element stiffness matrix in local
  coordinates is
• -since truss member connects two nodes, the
  transformation

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Coordinate transformations from local to global

• -element equation in local coordinate
• -transform to global
• -multiply both sides by T-1

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Global stiffness matrix
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Plane truss example

• Analyze the truss shown, given

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Plane truss example (continues) Member connects
nodes 1?j
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Plane truss example (continues)
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Plane truss example (continues)

75
Stress equation for PT

76
Space truss

• Better approximate reality
• Analogize torsion in RC structures
• Analogize arches and domes.

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Space truss

• 3D trusses

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Space truss
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Space truss global stiffness
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Example

• Analyze the truss shown, given

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ST example (solution) Member connects nodes 1?j

• member(1)1 to 2

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ST example (solution)

• member(2) 1 to 3

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ST example (solution)

• -member (3) 1 to 4

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ST example (solution)

• -combining equations

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ST example (solution)

• -equation of stress becomes

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ST example (solution)
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End of PT and ST analysis
Let Learning Continue
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• Ways of reducing DOFs

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1. Condensation

• Reducing any number of DOFs up-to 1
  DOF-equation.
• Example solve
• First step partition into number of DOFs that
  are needed to be removed, assume A, and those
  which are needed to remain, assume B.

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Condensation of matrices (continues)
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Example
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Example (continue)

• Hw use matrix condensation to solve (hint
  divide 2X2)

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2.Symmetry

• We only need to deal with half the structure
  around each symmetric plane. If two planes exist,
  we work with ¼.
• Example analyze if
• E20X103 MPa,

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Symmetry (continues)Notice 2 planes of symmetry

• -member 1 connects 1 to 5

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Symmetry (continues) Finding stresses
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Find reactions
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End of ways of reducing DOF's
Let Learning Continue
98
Finite element for beams
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Analysis of beam elements-forces

• -Analytical solution
• review mechanics of materials

100
Analysis of beam elements-deformations
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Analysis of beam elements
particular solution Homogeneous
solution
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Analysis of beam elements (cont.)analytical
solution example
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Analysis of beam elements (cont.)analytical
solution example (continued)
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General basic steps for finite element method
beam elements

• Input
• Problem to be solved
• Physics of problem
• Mathematical model

• Processing
• Propose theory
• Formulate equations
• Solve equations

Output 1. Verify compatibility 2. Verify
equilibrium 3. Verify stress-strain relationships
105
Input

• Problem analysis of the beam in the previous
  example
• Physics shear and moment deformations
• Mathematical model material, geometry and
  loading

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Processing 2.1 propose theory for solution

• a. Select element type beam, 1 element
• b. Select displacement functions
• v generic displacement

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2.1 propose theory for solution b. Select
displacement functions
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Shape Functions
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2.1 propose theory for solution c. Derive strain
and stress displacement relationships
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2.2 formulate equations

1. Derive element stiffness using principle of
   Virtual Work

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2.2 formulate equations a. Derive the element
stiffness matrix and equations
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2.2 formulate equations a. Derive the element
stiffness matrix and equations

• For a uniform downward load p0
• b. Transfer from local to global axes (No need)
• c. Assemble the element equations to obtain the
  global equations and introduce boundary
  conditions

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Example analytical solution
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Example FE solution structure one element
SIMILAR CONCLUSIONS AS BEFORE!
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Example FE solution discretize to two elements

• Let lL/2

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Example FE solution discretize to two elements
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Example FE solution discretize to two elements
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FE versus analytical
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End of fe for beams
Let Learning Continue
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• Finite Element Analysis of framed structures

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Plane framesprinciple of superposition
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Plane framesequivalent joint loads
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Plane framestransformations
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Plane framesglobal stiffness
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Example 1

• Given E18 GPa,
• A 0.05m2, I .0003m4
• Required analyze frame
• Solution
• Member 1 2 ? 1

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Example 1(continues)

• Member 2 3 to 2

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Example 1 (continues)
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Example 1(continues)

• Reactions
• Member 1 2 ? 1

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Example 1 (continues)
Member 2 3 ? 2
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Example 2

• Example- Repeat previous example neglecting
  axial deformations
• Member 1
• Member 2

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End of FE for frames
Let Learning Continue
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Finite Element Analysis of grid structures

• Dr. Abdul Razzaq Touqan
• Department of Civil Engineering

133
Analysis of torsional elements

• Analytical solution review mechanics of
  materials

134
Analysis of torsional elements (cont.)analytical
solution

• but
• Homogeneous solution particular solution

135
Analysis of torsional elements (cont.)analytical
solution example

• -analyze the following structure

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Analysis of torsional elements (cont.)analytical
solution example

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General basic steps for finite element method
applied here for grid elements

• Input
• Problem to be solved
• Physics of problem
• Mathematical model

• Processing
• Propose theory
• Formulate equations
• Solve equations

Output 1. Verify compatibility 2. Verify
equilibrium 3. Verify stress-strain relationships
140
Input

• Problem to be solved analysis of a shaft
  subjected to torsion loads
• Understand physics shaft will be subjected to
  shear stresses and torsional deformations
• Mathematical model define material, geometry and
  loading

141
Processing 1. propose theory for solution

• a. Select element type torsional element
  (2-node)
• b. Select displacement functions
• (1) f generic displacement

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2.1 propose theory for solution

• b. continues express u(x) in terms of nodal
  displacements using boundary conditions

143
2.1 propose theory for solution
c. Derive strain and stress displacement
relationships
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2.1 propose theory for solution
c. Derive strain and stress displacement
relationships
145
2.2 formulate equations

• a.Derive the element stiffness matrix and
  equations using principle of Virtual Work
  (general derivation)
• J Polar moment of inertia for torsional members

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2.2 formulate equationsa.Derive the element
stiffness matrix and equations

• For uniform torsion

147
2.2 formulate equations

• b. Transfer equations from local to global axes
  (later)
• c. Assemble the element equations to obtain the
  global equations and introduce boundary
  conditions

148
2.3 Solve equations

• a. Solve for the unknown DOF
• b. Solve for reactions
• c. Solve for resultant element strains and
  stresses

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3. Output

• Verify compatibility same as analytical
• Verify equilibrium reactiontL ok
• Verify stress strain relationship
• -resultant torque is constant unlike true
  solution which is linear

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Gridprinciple of superposition
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Gridequivalent joint loads
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Gridtransformations

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Gridglobal stiffness
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Example 1

Given Required analyze grid Solution
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Example 1(continues)

• Member 1

Member 2
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Example 1 (continues)
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Example 1 (continues)
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End of FE for grids
Let Learning Continue
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Discussion class

• Space frames

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Kjj

161
Kkj

162
Kkk

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Equivalent joint loads

164
Transformation matrix

165
End of FE for space frame
Let Learning Continue
166
Space-frame versus Shell

• Dr. Abdul Razzaq Touqan

167
1D Space-frame

168
2D -Shell

169
Methodology

1. Understand exact 1D
2. Perform analogical solutions between 1D and nD
   models
3. Build up experience with 3D models

170
Example

• Cantilever beam 3m span, 0.2m width by 0.3m depth
  made of concrete with weight density 2.5t/m and
  E2.5X10 t/m2. Find end span deflection and
  maximum stress at fixed end using
• Exact solution
• Finite element solution
• 1D space frame model
• 2D shell model
• 3D solid model

171
End of FE for 2D and 3D
Let Learning Continue
172

• Structural Modeling

173
Definition

174
Structural modeling assumptions

• Elements
• 1D
• 2D
• 3D
• Structures
• 1D structure with 1D elements
• 2D structure with 1D, 2D elements
• 3D structure with 1D, 2D, 3D elements

175
Loading assumptions

• Static
• Dynamic

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1D, 2D or 3D modeling 1D slab-beam-column
177
1D, 2D or 3D modeling 1D slab-beam-column/
continued
178
1D, 2D or 3D modeling 2D plane frame
179
1D, 2D or 3D modeling 3D space frame with
slabs/walls
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Laws versus theories

• Structural Analysis laws
• constitutive (stress-strain) relationships
• equilibrium equations essential
• compatibility equations optional/present
  challenge
• Structural Analysis theories
• Based on assumptions
• Assumptions based on available knowledge
• Available knowledge is constrained with available
  tools like hand calculators and personal
  computers
• Computer programs are based on assumptions on
  which the theoretical basis of the software was
  developed.

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Structural analysis system approach

• Input
• 1.Goal
• 2.Given
• 3.Create a mathematical model
• Processing
• 1.Propose a theory
• 2.Formulate equations
• 3.Solve the equations
• Output
• 1.Verify Compatibility
• 2. Verify equilibrium.
• 3. Verify stress-strain relationships.

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Structural analysis system approach input

• Input
• 1.Goal (ref creation versus man made),
  provide
• support system (science)
• functional system (facts)
• 2.Given (ref global versus local), understand
  physics of the system, i.e. specify system
  requirements
• strength, stiffness and stability
• function, freedom and form
• 3.Create a mathematical model (ref codes of
  practice), specify structural system that
  provide
• satisfaction
• fairness

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Structural analysis system approach processing

• Processing
• 1.Propose a theory reduce assumptions deviating
  model from reality\ref 3D nonlinear dynamic
  probabilistic soil-soil GIS interaction
• 2.Formulate equations according to state of
  knowledge and available tools \ref analytical,
  anatomical, analogical
• 3.Solve the equations analyze the structure\
  ref verify analysis laws

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Structural analysis system approach output

• Output
• 1.Verify Compatibility
• 2. Verify equilibrium.
• 3. Verify stress-strain relationships.

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example

• A single storey RC slab-beam factory structure
  shown next page
• Fixed foundations, 4 spans 5m bays in x and a
  single 8m span in y, 6m elevation
• E24GPa, µ0.2, ?2.5t/m3
• Slab 25cm thickness, drop beams 30cmX80cm,
  columns 30X60cm
• superimposed loads5kN/m2, live load9kN/m2
• Due to cracking of elements, assume modifiers
  for gross inertia
• Beam 0.35
• Column 0.7
• One way slab (0.35, 0.035)

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Analogical 1D analysis slab model

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1D analysis slab analysis
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1D analysis slab analysis

• Wd(.2524.55)11.125KN/m
• Wl9KN/m
• Wu1.211.1251.6927.75KN/m

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1D analysis slab analysis, values of bending
moment KN.m
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1D analysis slab analysis, values of reactions
in KN
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1D analysis beam and column analysis,

• Assume simply supported beam
• Beam C, Mu(1291.20.3.824.5)82 /81079
• Beam B, Mu(1591.20.3.824.5)82 /81319
• Beam A, Mu(54.521.20.3.824.5)82 /8492
• Column reactions
• Beam C, Ru (1290.3.824.5)8 /2540
• Beam B, Ru(1590.3.824.5)8 /2660
• Beam A, Ru (54.50.3.824.5)8 /2242

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Gravity equilibrium checks

• D
• Slab20X8X(0.25X24.55)1780KN
• Beams(5X82X20)X.8X.3X24.5470KN
• Columns10X6X.3X.6X24.5264.6KN
• Sum2514.6KN
• L
• R 20X8X91440KN

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Gravity equilibrium checks

• SAP results

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Homework

• Analyze and design a one story reinforced
  concrete structure (entertainment hall) made of
  one way solid slab sitting on drop beams
  supported on six square columns 50cm dimensions.
  The superimposed loads are 300kg/m2, and the live
  load equal to 400kg/m2
• Analyze using analogical (local practice
  slab-beam-column load path)
• Analyze using anatomical finite element 3D model
  (more actual representation)
• Compare analogical with anatomical

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End of structural modelings
Let Learning Continue