1. Chapter 11: Displacement Method of Analysis: Slope-Deflection Equations CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang

2. Force vs Displacement Methods • Force methods • Choose redundant forces • Use compatibility conditions or least work principle to solve these redundant forces • Displacement methods • Choose degrees of freedom (DOFs: displacement or rotation angles) • Relate internal forces to DOFs • Use equilibrium to solve DOFs • Obtain internal forces from DOFs By qB

3. Degrees of Freedom (DOFs) • Nodes, members (elements/components) and DOFs Nodes Member DOF qB qB C DC qC

4. Degrees of Freedom (DOFs)

5. Beam under Loading A B L A’ B’ MBA Assume moment/rotation + MAB • Choose degrees of freedom • Relate internal forces to DOFs • Use equilibrium to solve DOFs • Obtain internal forces from DOFs How do MAB & MBA relate to deformation at A & B? A B B’ A’

6. Slope-Deflection Relationship MAB1+MAB2 MBA1+MBA2 A A B B MBA B’ B’ MBA1 MBA2 A’ A’ MAB MAB1 MAB2 A B B’ A’ Moments to produce such a deformation Moments to resist loads without extra deformation From Table (Fixed end moments, FEM)

7. Slope-Deflection Relationship D L • Choose degrees of freedom • Relate internal forces to DOFs • Use equilibrium to solve DOFs • Obtain internal forces from DOFs A B MBA1 B’ A’ MAB1

8. Slope-Deflection Relationship A B MBA1 M1 D B’ L MAB1 A’ M2 B’ M2 M2 M1 M1 A’ + =

9. Slope-Deflection Relationship MAB1+MAB2 MBA1+MBA2 A A B B MBA B’ B’ MBA1 MBA2 A’ A’ MAB MAB1 MAB2 A B B’ A’ Moments to produce such a deformation Moments to resist loads without extra deformation From Table (Fixed end moments, FEM)

10. Slope-Deflection Relationship D A B L L A’ B’ • Choose degrees of freedom • Relate internal forces to DOFs • Use equilibrium to solve DOFs • Obtain internal forces from DOFs A B MBA B’ A’ MAB moment/rotation + =