Civil Engineering Reference
In-Depth Information
19.2 Factors
The analyses in this chapter and the next calculate ultimate limit states where the slope,
wall or foundation can be said to have collapsed. In practice engineers are required
to design safe and serviceable structures and to do this they apply factors to their
calculations. These factors are intended to move the design away from a collapse state
into a state in which there is no danger of collapse or where movements are acceptably
small. Similar procedures are employed throughout engineering design.
Some factors are factors of safety and they are intended to ensure that the structure
is not near its ultimate limit state. Other factors are load factors and they are intended
to ensure that the deformations remain within a small strain range to limit movements.
Factors of safety are not the same as load factors: each has a different purpose. Some-
times, instead of a single factor, partial factors are used. Each partial factor is applied
to a separate component of the calculation. Use of factors to ensure safety and to limit
movements were discussed in Chapter 18 and will be discussed further in later chapters
dealing with different structures.
19.3 Theorems of plastic collapse
In order to simplify stability calculations it is possible to ignore some of the conditions
of equilibrium and compatibility and to make use of important theorems of plastic
collapse. It turns out that, by ignoring the equilibrium condition, you can calculate
an upper bound to the collapse load so that if the structure is loaded to this value it
must collapse; similarly, by ignoring the compatibility condition you can calculate a
lower bound to the collapse load so that if the structure is loaded to this value it cannot
collapse. Obviously the true collapse load must lie between these bounds.
The essential feature of the upper and lower bound calculations is that rigorous
proofs exist which show that they will bracket the true collapse load. Thus, although
the two methods of calculation have been simplified by ignoring, for the first, equilib-
rium and, for the second, compatibility, no major assumptions are needed (other than
those required to prove the bound theorems in the first place). What has been lost by
making the calculations simple is certainty; all you have are upper and lower bounds
and you do not know the true collapse load (unless you can obtain equal upper and
lower bounds). Usually you can obtain upper and lower bounds that are fairly close
to one another so the degree of uncertainty is quite small.
I am not going to prove the plastic collapse theorems here and I will simply quote the
results. A condition required to prove the theorems is that the material must be perfectly
plastic. This means that, at failure, the soil must be straining at a constant state with
an associated flow rule so that the vector of plastic strain increment is normal to the
failure envelope (see Chapter 3). The first condition, straining at a constant state, is met
by soils at their ultimate or critical states, given by Eqs. (19.1) and (19.2). The second
condition is illustrated in Fig. 19.1(a) for undrained loading and in Fig. 19.1(b) for
drained loading.
In both cases elastic strains must be zero since the stresses remain constant; thus
total and plastic strains are the same. For undrained loading the failure envelope given
by Eq. (19.1) is horizontal and the volumetric strains are zero (because undrained
means constant volume) and so the vector of plastic strain
p
is normal to the failure
δε
