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?
Degree of Freedom
→ + +
+ +
Y
Z
X
+
←
qY
Y
Z
X Y
Z
X
dx Y
Z
X
dy Y
Z
X
dz Y
Z
X
qX
Y
Z
X
qz
Y
Z
X
▶ If only in-plane motion available
Degree of Freedom
X Y
dx
dy qZ
Z
In-plane
deformation dx
Y
Z
X dy
Y
Z
X
qz
Y
Z
X
qY
Y
Z
X
dz
Y
Z
X
qX
Y
Z
Out-of-plane X
deformation
▶ Plane frame
Idealization and Number of DOF
X Y
Z
X
dy
dz
dx
qy
qz x
q
dx dy
qz
▶ Idealization of a plane frame
Idealization and Number of DOF
(a) (b) (c)
(d) (e)
▶ 1-DOF vs. Multi-DOF
Choices in Structural Modeling
Horizontal 1-DOF Vertical 1-DOF 2-DOF Multi-DOF 1-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.
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.
Virtual Displacement Principle for a Rigid Body
Virtual Displacement Principle for a Rigid Body
Virtual Displacement Principle for a Rigid Body
▶ Example 2. The loading and bending moment diagram for a frame are shown. Find the magnitude of load F.
(1) By Equilibrium Equations
Virtual Displacement Principle for a Rigid Body
(2) By Virtual work on Equivalent Mechanism
Virtual Displacement Principle for a Rigid Body
§ 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
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=0
True 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.
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)
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.
