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

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

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

Degrees of Freedom (DOFs)

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?

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)

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

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

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

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