STATICALLY INDETERMINATE PROBLEMS I

1
STATICALLY INDETERMINATE PROBLEMS I

• How do we obtain indeterminate reactions when the
  number of equilibrium equations is less then the
  number of support forces?

2
The Role of Deformability in Statical
Indeterminacy

• 1. Rigid supports deformable beam

3
The Role of Deformability in Statical
Indeterminacy

• 2. Rigid beam elastic supports

4
Equilibrium of Rigid Bodies with Lumped Elastic
Supports

• 1. Rigid body equilibrium
• 2. Kinematics of infinitesimal rigid body
  displacements
• 3. Lumped models for supports e.g. linear and
  torsional springs

5
Kinematics of Infinitesimal Rigid Body
Displacements

• Translation
• All points in the rigid body have the same
  constant displacement vector

6
Kinematics of Infinitesimal Rigid Body
Displacements

• Rotation about a fixed point
• Displacement vector has magnitude rw with
  direction perpendicular to line joining fixed
  point and point of interest.

Rotation vector wwk
Displacement vector uw?r
7
Kinematics of Infinitesimal Rigid Body
Displacements

• General rigid body motion
• Translation of a point and rigid rotation about
  an axis through that point.

Displacement vector uu0 w?r
8
Cartesian Components Plane Case
u u0 w?r

• Formula
• Intuitive derivation

9
Lumped Support Models

• Linear springs
• Torsional springs

10
The Statically Indeterminate Simply Supported Beam

• Free Body Diagram

11
The Indeterminate Beam

• Governing Equations

12
The Indeterminate Beam

• Force Constraint Equation

13
The Indeterminate Beam

• Governing Equations

3 equations for 3 unknowns!
14
The Indeterminate Beam

• Solution
• Non-dimensional variables
• Solution

15
The Indeterminate Beam
Does the addition of a support at A always cause
a decrease in the support forces at B and C?
NO!!!