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

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 ByBy BB

3. Degrees of Freedom (DOFs) Nodes, members (elements/components) and DOFs BB BB Nodes Member DOF C CC CC

4. Degrees of Freedom (DOFs)

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

6. Slope-Deflection Relationship AB A’ B’ M AB M BA A B A’ B’ M AB1 M BA1 Moments to produce such a deformation Moments to resist loads without extra deformation A B A’ B’ M AB2 M BA2 M AB1 +M AB2 M BA1 +M BA2 From Table (Fixed end moments, FEM)

7. Slope-Deflection Relationship AB A’ B’ M AB1 M BA1 L  Choose degrees of freedom Relate internal forces to DOFs Use equilibrium to solve DOFs Obtain internal forces from DOFs

8. Slope-Deflection Relationship M1M1 M2M2 M1M1 M2M2 + = M AB1 AB A’ B’ M BA1 L  M1M1 M2M2 A’ B’

9. Slope-Deflection Relationship Moments to produce such a deformation Moments to resist loads without extra deformation A B A’ B’ M AB2 M BA2 M AB1 +M AB2 M BA1 +M BA2 A B A’ B’ M AB1 M BA1 From Table (Fixed end moments, FEM) AB A’ B’ M AB M BA

10. Slope-Deflection Relationship AB A’ B’ M AB M BA L  A A’ B B’ L = moment/rotation + Choose degrees of freedom Relate internal forces to DOFs Use equilibrium to solve DOFs Obtain internal forces from DOFs