mu
ductility capacity
mu = umax / uy
ductility of different building types
- well detailed reinforced concrete and steel buildings: ‘ductile’
- unreinforced masonry buildings: ‘brittle’
mu = 1
implies linear response - relates to zero structural damage
mu = 10
implies structural collapse
relationship of ductility and load capacity
ductile buildings can sustain larger loads than brittle buildings
two parts to hysteresis curve
- backbone or virgin loading curve
- unloading and reloading curve
‘backbone’ or ‘virgin loading’ curve
describes response of the material when subject to monotonic loading
unloading and reloading rules
specify how the material responds when it is unloaded/reloaded
two methods to predict non-linear dynamic response of a system
- solve equation of motion incrementally
- use empirical relationships between elastic spectral demands and inelastic response parameters
Errors using tangent approximation to secant stiffness
- tangent and secant stiffness differ greatly of a single time step
- form cumulative errors over multiple time steps
modified newton-raphson method
minimises errors by using an iterative procedure
- continue process until additional increments in displacement and/or residual force become acceptably small
Original newton-raphson method
tangent recalculated at each iteration
- converges with fewer iterations
- BUT takes MUCH longer
Ry
force reduction factor
- defines how much smaller the maximum inelastic force compared to the maximum elastic force
Ry = f0 / fy
Cy
yield strength coefficient
- independent of elastic spectral demands - purely a characteristic of the building
Cy = Fy / mg
Ry - mu - T relationships
- obtained by analysing a large number of SDOF systems to ground motions
- can use relationships to relate inelastic and elastic spectral demands
two important trends when looking at umax/u0 vs T
- ratio umax/u0 broadly classified into two regions
- moderate/long periods (T >0.4) inelastic and elastic disp, are similar
- short periods, inelastic>elastic and ratio generally increases with decreasing T
- moderate/long T, umax/u0 insensitive to Ry
- short periods, umax/u0 increases with increasing Ry
Equal displacement approximation
for moderate to long period structures
- inelastic and elastic disp similar (0.5
Equal Energy approximation
short periods
Sensitivity of R - mu - T
- hysteretic models with larger/fatter loops tend to have lower ratios
- lower magnitude events tend to generate longer ground motions
P-delta effects
Fy,pdelta = Fy - ( P x delta / h)
what happens when you consider P-delta
- tends to reduce equivalent lateral forces a structure can resist
- reduction in apparent system stiffness
- increase in system period of vibration
- reduce apparent acceleration demands
- increase apparent displacement demands, therefore typically non conservative to ignore P-delta
