Structural Dynamics / Nonlinear structural dynamics
Ductility, and what it buys
Strength reduction factors, the equal-displacement and equal-energy rules, equivalent damping, and the demand that peak displacement does not describe.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.
- Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
- Independent nonlinear engineering review needed for constitutive calibration, cyclic degradation, P–Δ effects, member/system ductility and post-event residual definition. R/q factors cannot be selected directly from a member ductility number.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
A yielding structure can resist an earthquake through inelastic deformation, provided its system and details have adequate capacity. This lesson compares idealised strength-reduction relations, displacement demand and residual response.
A code behaviour factor involves more than a member ductility value. Its use depends on the structural system, overstrength, detailing and the applicable design provisions.
By the end of this lesson you should be able to
- Apply the equal-displacement and equal-energy approximations with explicit assumptions and limits
- Convert between required strength, ductility demand and displacement demand
- Explain the steady-cycle basis and calibration limits of equivalent viscous damping
- Treat residual displacement as a separate demand from peak displacement
What you should already know
- Hysteresis and the bilinear model (Lesson 16.1)
- Response spectra (Module 14)
- Energy dissipation per cycle (Module 7)
Strength reduction and ductility
The equal-displacement approximation assumes that an inelastic oscillator reaches approximately the displacement of an elastic oscillator with the same initial stiffness. With R = felastic/fy and μ = umax/uy, this assumption gives R = μ.
An equal-energy idealisation equates the areas under elastic and elastic–perfectly plastic monotonic force–displacement curves. It gives R = √(2μ−1). At μ = 4 the two estimates are 4 and 2.65 respectively.
Their usefulness depends on the period, excitation, hysteresis model and strength level. There is no universal 0.5 s switch. The equal-energy formula is independent of period and therefore cannot itself describe the very-short-period limit R → 1. Use a justified inelastic spectrum or the applicable assessment method for that range.
Displacement demand
Under the equal-displacement assumption, reducing yield strength leaves the estimated peak displacement unchanged and increases ductility demand. That is an approximation, not a guarantee for every record. A member ductility capacity cannot automatically be assigned as a global reduction factor; the load path, mechanism, degradation and stability also matter.
Reduced-force elastic-analysis displacements must be interpreted using the displacement rules of the applicable design method. Movement-sensitive details need the relevant local or relative displacement demand.
Equivalent viscous damping
For a stable symmetric elastic–perfectly plastic cycle, using the peak secant strain energy gives ζeq = (2/π)(1−1/μ). This yields 31.8% at μ = 2 and 47.7% at μ = 4.
The equality applies to the chosen steady cycle. Earthquake cycles vary in amplitude, so substituting that value into a linear model needs calibration for the record and hysteresis model. It is not a universal effective damping ratio or a fixed percentage overestimate.
Residual displacement
A yielding system may retain a permanent offset after motion has decayed. The last displacement in a finite record can still contain free vibration; extend the analysis sufficiently before calling it a settled residual.
Residual response can be sensitive to loading sequence and constitutive assumptions. Assess an appropriate record set using a justified statistic and check convergence. Positive post-yield stiffness affects this response but kinematic hardening does not guarantee self-centring. Repairability also needs project-specific deformation and damage criteria.
Choosing the analysis
Nonlinear analysis is useful where the question involves yielding, contact, changing geometry, damage sequence or residual response. It may also be needed for a serviceability problem with nonlinear behaviour. Choose a model that answers the question, state uncertainty in its parameters and test sensitivity. Additional numerical precision does not remove uncertainty in the physical model.
Worked example
A stated equal-displacement comparison
Given
- A steel moment frame, elastic period T₁ = 0.90 s
- Elastic spectral acceleration at that period: 0.62 g
- Seismic mass 240 tonnes
- For this hypothetical system-level comparison, assume μ = 4 is available; actual capacity remains unverified
Find
The force, ductility and displacement implied by the stated equal-displacement approximation. This is not a code design or capacity verification.
Assumptions
- Idealised SDOF representation; equal displacement is assumed for this example, not established by the period alone
- System-level deformation capacity is assumed for arithmetic only and needs separate verification
Predict first
Assume equal displacement for two idealised systems with the same mass, initial stiffness and period of 1.2 s, one elastic and one with R = 4. What does that assumption predict for their peak displacements?
Practice
A structure has an elastic base shear demand of 1 460 kN and is designed with a behaviour factor of 4. What is the design base shear, in kN?
Practice
An elastic–perfectly plastic system reaches a ductility of 4. What equivalent viscous damping ratio does the energy-equivalence formula ζeq = (2/π)(1 − 1/μ) give, as a percentage?
Practice
The same formula at a ductility of 2 gives what equivalent damping ratio, as a percentage?
Practice
An idealised frame yields at 31.2 mm and reaches μ = 4. What peak displacement does that imply, in millimetres? Local cladding movement demands require separate assessment.
Check yourself
Which expression follows from equating monotonic elastic and elastic–perfectly plastic energy areas?
Check yourself
Two nonlinear analyses of the same building under different records give peak displacements within 5% of each other but residual displacements differing by a factor of three. What should be concluded?
Worked example
What a ductility of four is worth
Given
- A structure designed to reach a displacement ductility of 4
- Two candidate rules: equal displacement and equal energy
Find
The strength reduction each rule permits
Worked example
Reading a secant period off a hysteresis loop
Given
- A structure with an elastic period of 0.6 s
- It is pushed to a ductility of 4, behaving elastic–perfectly plastic
Find
The effective period at peak displacement
Summary
- R = μ and R = √(2μ−1) follow from different idealisations; neither has a universal period boundary
- A behaviour factor reduces strength, not displacement — drift must be amplified before it is checked
- Steady-cycle equivalent damping needs calibration before use with earthquake response
- Residual displacement is a separate demand, badly conditioned, and needs a record set
- Post-yield stiffness influences residual response but does not guarantee self-centring
- Nonlinear analysis earns its cost when ductility demand, concentration or residuals are the question