In the context of earthquake engineering and structural dynamics, mode shapes represent the characteristic deformation patterns that structures undergo at specific natural frequencies during free vibration. These shapes provide vital insights into how structures respond dynamically and are central to modal analysis — a core concept in evaluating seismic behavior. Understanding mode shapes is crucial for designing earthquake-resistant structures and ensuring that resonance or amplified motions do not result in catastrophic failure.

Free vibration refers to the vibration of a system without any external force, after an initial disturbance. In multi-degree-of-freedom (MDOF) systems, the vibration response is a combination of several independent modes of vibration, each with a unique natural frequency and mode shape.
• Definition: A mode shape is the deformation pattern of a structure at a specific natural frequency during free vibration.
• Each mode shape corresponds to one natural frequency.
• The number of possible mode shapes is equal to the number of degrees of freedom (DOFs) in the system.
Example: For a 3-DOF shear building, three distinct mode shapes will exist, each representing a different vertical displacement profile across the floors.

Consider an undamped linear MDOF system governed by:
[M]{u¨}+[K]{u}={0}
Where:
• [M] = Mass matrix
• [K] = Stiffness matrix
• {u} = Displacement vector
Assuming a harmonic solution of the form:
{u(t)}={ϕ}sin(ωt)
Substituting and simplifying leads to the eigenvalue problem:
([K]−ω²[M]){ϕ}={0}
• ω = natural frequency
• {ϕ} = mode shape (eigenvector)
This formulation yields eigenvalues (natural frequencies squared) and eigenvectors (mode shapes).

Mode shapes are orthogonal with respect to both the mass and stiffness matrices:
• Mass orthogonality:
{ϕ}^T[M]{ϕ}=0 for i≠j
• Stiffness orthogonality:
{ϕ}^T[K]{ϕ}=0 for i≠j
Orthogonality is key in modal superposition and decoupling the equations of motion.
Mode shapes are not unique in magnitude. They can be normalized for analytical convenience:
• Mass normalization:
{ϕ}^T[M]{ϕ}=1
• Stiffness normalization:
{ϕ}^T[K]{ϕ}=ω²

Mode shapes can be computed by solving the eigenvalue problem using:
• Analytical methods for small systems (e.g., 2-DOF, 3-DOF)
• Numerical methods for large systems using:
– Subspace iteration
– Lanczos algorithm
– Rayleigh-Ritz method
For high-rise or complex structures, software like SAP2000, ETABS, or ANSYS is commonly used.

15.5.1 First Mode Shape
• Usually involves global movement of the entire structure.
• Dominant in seismic analysis due to its lower frequency and higher participation.
15.5.2 Higher Mode Shapes
• Represent localized or complex motion.
• Become significant in irregular or tall structures.
• Often show curvatures, torsions, or out-of-phase displacements between different parts.

• Seismic Response: Structures tend to vibrate in their natural modes during seismic excitation.
• Design Optimization: Identifying mode shapes helps in modifying geometry or stiffness to improve performance.
• Modal Participation Factor: Indicates how much each mode contributes to the overall response.
• Time History and Response Spectrum Analysis: Use mode shapes to determine maximum displacements and internal forces.
• Mode Combination Rules: Use of SRSS (Square Root of Sum of Squares) or CQC (Complete Quadratic Combination) for combining modal responses.

• Mode shapes help identify weak stories, soft floors, and critical joints.
• Used in design of tuned mass dampers (TMDs), base isolation systems, and retrofit schemes.
• Changes in mode shapes before and after retrofitting provide insight into structural improvement.