457.649 Advanced Structural Analysis Part I: Fundamentals

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


▶ What is the degree of freedom?

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Degree of Freedom

+→ +

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←Yq

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xd

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yd

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▶ If only in-plane motion available

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Degree of Freedom

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In-plane deformation xd

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Xyd

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zq

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YqY

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XOut-of-plane deformation


▶ Plane frame

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Idealization and Number of DOF

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yd

zdxd

yq

zqxq

xd

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▶ Idealization of a plane frame

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Idealization and Number of DOF

(a) (b) (c)

(d)(e)


▶ 1-DOF vs. Multi-DOF

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Choices in Structural Modeling

Horizontal 1-DOF Vertical 1-DOF 2-DOF Multi-DOF1-DOF


§ 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


▶ 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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Seoul National UniversityStructural Design Laboratory8


Seoul National UniversityStructural Design Laboratory9



▶ 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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(2) By Virtual work on Equivalent Mechanism

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§ 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


▶ 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


§ 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


▶ 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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457.649 Advanced Structural Analysis Part I: Fundamentals