Displacement methods
What is Displacement methods?
The moment required to produce a unit rotation at the end of a member, assuming the other end is fixed.
Key formula / rule: Slope-Deflection Equation (Prismatic Member)
Key points
- Understand the fundamental principles of displacement methods.
- Apply the slope-deflection method to analyze beams and frames.
- Apply the moment distribution method to analyze indeterminate structures.
- Formulate and solve equilibrium equations for joint displacements.
Common exam trap
Incorrectly defining sign conventions for moments and displacements.
Definitions
- Term
Stiffness Factor (K)
- Meaning
The moment required to produce a unit rotation at the end of a member, assuming the other end is fixed.
- Term
Distribution Factor (DF)
- Meaning
The proportion of an unbalanced moment at a joint that is distributed to a particular member connected to that joint.
- Term
Carry-over Factor (COF)
- Meaning
The ratio of the moment transferred to the far end of a member to the moment applied at the near end, assuming the far end is fixed.
- Term
Fixed-end Moments (FEM)
- Meaning
Moments that occur at the ends of a member when its ends are restrained against rotation and translation, and loads are applied to the member.
Learning objectives
Understand the fundamental principles of displacement methods.
Apply the slope-deflection method to analyze beams and frames.
Apply the moment distribution method to analyze indeterminate structures.
Formulate and solve equilibrium equations for joint displacements.
Calculate member forces and reactions from joint displacements.
Formulae
- Name
Slope-Deflection Equation (Prismatic Member)
- Note
MAB: Moment at end A of member AB. θA, θB: Rotations at ends A and B. Δ: Relative settlement of end B with respect to A. L: Length of member AB. E: Young's Modulus. I: Moment of Inertia. FEMAB: Fixed-end moment at A due to applied loads.
- Expression
MAB = (2EI/L)(2θA + θB - 3Δ/L) + FEMAB
- Name
Stiffness Factor (Prismatic Member)
- Note
Stiffness factor for a member resisting rotation at one end while the other is fixed.
- Expression
K = EI/L
- Name
Distribution Factor (DF)
- Note
DFij is the fraction of an applied moment at joint 'j' that is resisted by member 'ij'. ΣKik is the sum of stiffness factors of all members connected at joint 'j'.
- Expression
DFij = Kij / ΣKik
- Name
Carry-over Factor
- Note
For prismatic members, half of the moment applied at one end is carried over to the other fixed end.
- Expression
COF = 0.5
Prerequisites
Statics (Equilibrium of forces and moments)
Mechanics of Materials (Stress-strain, bending stress, shear stress)
Basic concepts of indeterminate structures
Sign conventions in structural analysis
Common mistakes
Incorrectly defining sign conventions for moments and displacements.
Errors in calculating stiffness and carry-over factors.
Misapplication of boundary conditions (e.g., fixed, pinned, roller supports).
Algebraic errors when solving simultaneous equations.
Keywords
Displacement Method
Slope-Deflection
Moment Distribution
Stiffness
Indeterminate Structures
Joint Equilibrium
Rotation
Deflection
Carry-over Factor
Distribution Factor
Practice preview
Which of the following is the primary unknown in displacement methods of structural analysis?…
easy
Displacement methods of structural analysis are based on satisfying which of the following conditions?…
easy
Which of the following is an example of a displacement method of structural analysis?…
easy