A mode shape is the characteristic deformation pattern of a structure when vibrating at one of its natural frequencies. Each mode shape describes the relative displacement of all points in the structure and is fundamental to modal analysis and response spectrum analysis.
A mode shape (Ο) is the spatial pattern of deformation that a structure assumes when vibrating freely at one of its natural frequencies. For a multi-degree-of-freedom (MDOF) structure, there are as many mode shapes as there are degrees of freedom, each associated with a specific natural frequency. The first mode (fundamental mode) typically has the longest period and the simplest shape β for a building, it is usually a smooth curve that increases with height. Higher modes have shorter periods and more complex shapes, with nodes (points of zero displacement) that divide the structure into regions of opposite motion. The mode shapes and natural frequencies together form the modal properties of the structure, determined by solving the eigenvalue problem KΟ = ΟΒ²MΟ, where K is the stiffness matrix and M is the mass matrix.
Mode shapes are the foundation of modal analysis, the primary tool for understanding how structures respond to dynamic loads. Any deformed shape of a linear structure can be expressed as a weighted sum of its mode shapes, with the weights determined by the distribution of the applied load and the modal participation factors. In earthquake engineering, this decomposition allows the response of a complex MDOF structure to be computed as the sum of responses of simpler SDOF systems β one per mode β using the response spectrum. This is the basis of modal response spectrum analysis, the most widely used dynamic analysis method in seismic design practice. Typically, only the first few modes contribute significantly to the response, especially for regular structures, though irregular or tall structures may require many modes to capture the full response accurately.
Mode shapes are determined analytically, numerically, or experimentally. Analytical solutions exist for simple structures (uniform beams, shear buildings), while numerical eigenvalue analysis of finite element models is used for complex structures. Experimentally, mode shapes are identified from measurements of structural response under ambient, forced, or seismic excitation β using techniques such as frequency domain decomposition (FDD), stochastic subspace identification (SSI), or eigensystem realization algorithm (ERA). The identified mode shapes provide a direct validation of the analytical model and are the basis for structural health monitoring, since damage typically alters the mode shapes in characteristic ways. In particular, damage localization methods often rely on changes in the curvature of mode shapes or in the modal strain energy distribution, which are more sensitive to local damage than changes in natural frequencies alone.