Design Point GeometryNormalised load effect and resistance: the reliability index β as the shortest distance to the limit state · CIEM5000 Safety & Reliability
Plane view
3D view · drag to rotate
What the figures show
Dividing E by σE and R by σR makes the contours of equal probability circles around the mean point M. One ring is one standard deviation.
The limit state E = R becomes the line E/σE = (σR/σE) · R/σR, through the origin with slope σR/σE (45° only if σR = σE). Above the line E > R: failure.
β is the shortest distance from M to that line, in standard deviations. The point where the dashed circle of radius β touches the line is the design point P: the most likely combination of E and R at failure.
The projections of the arrow M → P are the contributions αRβ (resistance side) and −αEβ (load side). Their squares add up: αR2 + αE2 = 1.
In 3D: the hill is the joint probability density. The orange floor area is the failure region; the probability of failure is the tiny volume of hill on that side of the wall.