457.649 advanced structural analysis part i: fundamentals
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Seoul National UniversityStructural Design Laboratory
Structural Design Lab.(Prof. Ho-Kyung Kim)Dept. of Civil & Environmental Eng.
Seoul National University
457.649 Advanced Structural Analysis
Part I: Fundamentals of displacement method
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Seoul National UniversityStructural Design Laboratory
▶ What is the degree of freedom?
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Degree of Freedom
+→ +
++
Y
Z
X
+
←Yq
Y
Z
X
Y
Z
X
xd
Y
Z
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yd
Y
Z
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zd
Y
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Xq
Y
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zq
Y
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X
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Seoul National UniversityStructural Design Laboratory
▶ If only in-plane motion available
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Degree of Freedom
X
Y
xd
yd
Zq
Z
In-plane deformation xd
Y
Z
Xyd
Y
Z
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zq
Y
Z
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YqY
Z
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zd
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Xq
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XOut-of-plane deformation
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Seoul National UniversityStructural Design Laboratory
▶ Plane frame
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Idealization and Number of DOF
X
Y
Z
X
yd
zdxd
yq
zqxq
xd
yd
zq
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Seoul National UniversityStructural Design Laboratory
▶ Idealization of a plane frame
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Idealization and Number of DOF
(a) (b) (c)
(d)(e)
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Seoul National UniversityStructural Design Laboratory
▶ 1-DOF vs. Multi-DOF
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Choices in Structural Modeling
Horizontal 1-DOF Vertical 1-DOF 2-DOF Multi-DOF1-DOF
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Seoul National UniversityStructural Design Laboratory
§ Any rigid body§ In equilibrium§ Resultant force = 0§ Resultant couple = 0
§ Hence, if given a small translational or rotational displacement:§ WE=Σ(Work done by R1, R2, …, R5) = Work done by resultant force = 0
Where WE: external work = work done by external forces.Later will consider work by internal forces.
▶ Point to note§ External forces on body must be in equilibrium.§ Displacement of body must be small – otherwise may not remain in equilibrium.§ Displacement is introduced for mathematical purposes only – i.e. it is an imaginary,
or virtual displacement.§ The forces are given - Hence real.
We create the displacement – Hence imaginary.We make the displacement small.
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Virtual Displacement Principle for a Rigid Body
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Seoul National UniversityStructural Design Laboratory
▶ Extension to Mechanism§ If principle applies to a single rigid body, it also applies to a number of rigid bodies to
form a mechanism.
▶ Example 1. Find: RC; MD; VD
Procedure: 4 steps(1) Create a mechanism which is in equilibrium. The beam is statically determinate. To
create a mechanism we must introduce one or more “releases”.(2) Identify the real forces acting on the mechanism. Remember: the mechanism must
be in equilibrium.(3) Introduce a small(imaginary) displacement of the mechanism. Calculate the
displacements for each of the forces acting on the mechanism.(4) Write out and solve the work equations WE = 0.
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Virtual Displacement Principle for a Rigid Body
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Virtual Displacement Principle for a Rigid Body
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Virtual Displacement Principle for a Rigid Body
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Seoul National UniversityStructural Design Laboratory
▶ Example 2. The loading and bending moment diagram for a frame are shown. Find the magnitude of load F.
(1) By Equilibrium Equations
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Virtual Displacement Principle for a Rigid Body
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Seoul National UniversityStructural Design Laboratory
(2) By Virtual work on Equivalent Mechanism
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Virtual Displacement Principle for a Rigid Body
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Seoul National UniversityStructural Design Laboratory
§ Structure in Equilibrium, as previously defined.§ Impose small, imaginary and compatible displacement, as previously defined. Nodes
displace as rigid bodies. Elements undergo rigid body displacement plus deformation.§ WE = work done by external forces moving through corresponding (node)
displacement.§ WI = work done by element actions moving through element deformations.§ WE=WI
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Virtual Displacement Principle for a Deformable Body
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Seoul National UniversityStructural Design Laboratory
▶ Proof:(a) All joints are in equilibrium as rigid bodies.
Hence work by R’s +work by S’s on nodes=0True for any single node, hence for all nodes taken together.
(b) Forces S on elements are equal and opposite to forces S on nodes, and displacements of nodes, and element ends are same.
Hence work by S on nodes = - (work by S on elements)Work by R on nodes = WE(external)
Work by S on elements = WI(internal)Hence WE + (-WI)=0 or WE = WI
(c) Important to note that in calculating WI we can consider element deformations only - no need to consider rigid body displacements.
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Virtual Displacement Principle for a Deformable Body
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Seoul National UniversityStructural Design Laboratory
§ In proceeding proof: forces and actions were REAL displacement and deformation were IMAGINARY.
§ But exactly the same proof applies if: • Forces and actions are IMAGINARY.• Displacements and deformations are REAL.
§ This is the virtual forces principle.
▶ Requirements(1) A REAL displacement – deformation system which is:
(a) Compatible(b) Small ← note
(2) An IMAGINARY force-action system which is:(a) In equilibrium
WE* = WI
* (use * to distinguish form virtual displacements principle)
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Virtual Forces Principle
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Seoul National UniversityStructural Design Laboratory
▶ Geometry(Kinematics) of small angles and displacements
▶ Read carefully pp.420-428, pp.246-250 in “Elementary Structural Analysis, 4th Ed.” by Norris et al.
▶ Read carefully Chapter 3 in “Computer-Assisted Structural Analysis and Modeling” by Hoit.
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