Size effect on structural strength

The RVE is here defined as the smallest material volume whose failure suffices to make the whole structure fail.

For most normal-scale applications to metals and fine-grained ceramics, except for micrometer scale devices, the size is large enough for the Weibull theory to apply (but not for coarse-grained materials such as concrete).

Another check is that the histogram of the strengths of many identical specimens must be a straight line when plotted in the Weibull scale.

This kind of size effect represents a transition between two power laws and is observed in brittle heterogenous materials, termed quasibrittle.

These materials include concrete, fiber composites, rocks, coarse-grained and toughened ceramics, rigid foams, sea ice, dental ceramics, dentine, bone, biological shells, many bio- and bio-inspired materials, masonry, mortar, stiff cohesive soils, grouted soils, consolidated snow, wood, paper, carton, coal, cemented sands, etc.

On the micro- or nano scale, all the brittle materials become quasibrittle, and thus must exhibit the energetic size effect.

A simple intuitive justification of this size effect may be given by considering the flexural failure of an unnotched simply supported beam under a concentrated load

, and so (according to the first two terms of the binomial expansion) one may approximate it as which is the law of Type 1 deterministic size effect (Fig.

, for which the foregoing argument does not apply; and (b) to satisfy the asymptotic condition that the deterministic size effect must vanish for

5 for a general structural geometry has been given by applying dimensional analysis and asymptotic matching to the limit case of energy release when the initial macro-crack length tends to zero.

2d) shows a comparison of the last formula with the test results for many different concretes, plotted as dimensionless strength

Analysis of the multiscale transition to the material macro-scale then shows that the RVE strength distribution is Gaussian but with a Weibull (or power-law) left tail whose exponent

Note that the finiteness of the chain in the weakest-link model captures the deterministic part of size effect.

4b), or for structures in which a large crack, similar for different sizes, forms stably before the maximum load is reached.

Because the location of fracture initiation is predetermined to occur at the crack tip and thus cannot sample the random strengths of different RVEs, the statistical contribution to the mean size effect is negligible.

Such behavior is typical of reinforced concrete, damaged fiber-reinforced polymers and some compressed unreinforced structures.

at the tip, relieves the stress, and thus also the strain energy, from the shaded undamaged triangles of slope

8 is applicable in general, and that the dependence of its parameters on the structure geometry has approximately the following form: where

= dimensionless energy release function of linear elastic fracture mechanics (LEFM), which brings about the effect of structure geometry;

data from tests of geometrically similar notched specimens of very different sizes is a good way to identify the

Numerical simulations of failure by finite element codes can capture the energetic (or deterministic) size effect only if the material law relating the stress to deformation possesses a characteristic length.

This was not the case for the classical finite element codes with a material characterized solely by stress-strain relations.

One simple enough computational method is the cohesive (or fictitious) crack model, in which it is assumed that the stress

All these computational methods can ensure objectivity and proper convergence with respect to the refinement of the finite element mesh.

However, the fractal properties have yet not been experimentally documented for a broad enough scale and the problem has not yet been studied in depth comparable to the statistical and energetic size effects.

Taking the size effect into account is essential for safe prediction of strength of large concrete bridges, nuclear containments, roof shells, tall buildings, tunnel linings, large load-bearing parts of aircraft, spacecraft and ships made of fiber-polymer composites, wind turbines, large geotechnical excavations, earth and rock slopes, floating sea ice carrying loads, oil platforms under ice forces, etc.

Such statistical information, underlying the safety factors, is obtainable only by proper extrapolation of laboratory tests.

The safety factors, of course, give large safety margins—so large that even for the largest civil engineering structures the classical deterministic analysis based on the mean material properties normally yields failure loads smaller than the maximum design loads.

Such high failure probability is intolerable as it adds significantly to the risks to which people are inevitably exposed.

In fact, the historical experience shows that very large structures have been failing at a frequency several orders of magnitude higher than smaller ones.

Remnants of the Malpasset Dam in Maritime Alps, France, which failed at its first filling in 1959 and caused a giant flood that wiped out the town of Frejus, with several hundred fatalities. This dam, the tallest and slenderest at that time, failed because of an excessive horizontal slip in the gneissic abutment. The tolerable displacement considered in design is not known but, if calculated today, the size effect would reduce it to about one half of the value according to the design procedures in the 1950s.
Fig. 1: Size Effect on Structural Strength
Fig. 2
Fig. 4
Fig. 5 Schematic explanation of failure of Sleipner A Oil Platform, Norway 1991. The tricell of this $500 million, 190m meters tall, structure imploded under water head of 67 m, causing the platform so sink within 18 minutes (no fatalities). A government commission identified two factors causing failure: poor detailing of reinforcement, and poor finite element mesh. A separate investigation documented a third contributing factor: the size effect in the shear failure shown, which reduced the shear capacity by about 40%.