The ultimate load is reached when these conditions cease to exist and thus the struc-
ture collapses.
Thus, elastic design is concerned with an allowable stress, which equals the yield-
point stress divided by an appropriate factor of safety. In contrast, plastic design is con-
cerned with an allowable load, which equals the ultimate load divided by an appropriate
factor called the load factor. In reality, however, the distinction between elastic and plas-
tic design has become rather blurred because specifications that ostensibly pertain to elas-
tic design make covert concessions to plastic behavior. Several of these are underscored
in the calculation procedures that follow.
In the plastic analysis of flexural members, the following simplifying assumptions are
made:
- As the applied load is gradually increased, a state is eventually reached at which all
fibers at the section of maximum moment are stressed to the yield-point stress, in ei-
ther tension or compression. The section is then said to be in a state of plastification. - While plastification is proceeding at one section, the adjacent sections retain their lin-
ear-stress distribution.
Although the foregoing assumptions are fallacious, they introduce no appreciable
error.
When plastification is achieved at a given section, no additional bending stress may be
induced in any of its fibers, and the section is thus rendered impotent to resist any incre-
mental bending moment. As loading continues, the beam behaves as if it had been con-
structed with a hinge at the given section. Consequently, the beam is said to have devel-
oped a plastic hinge (in contradistinction to a true hinge) at the plastified section.
The yield moment My of a beam section is the bending moment associated with initial
yielding. The plastic moment Mp is the bending moment associated with plastification.
The plastic modulus Z of a beam section, which is analogous to the section modulus
used in elastic design, is defined by Z = MJf^ where/J, denotes the yield-point stress. The
shape factor SF is the ratio of Mp to My9 being so named because its value depends on the
shape of the section. Then SF = MJMy =fyZ/(fyS) = Z/S.
In the following calculation procedures, it is understood that the members are made of
A36 steel.
ALLOWABLE LOAD ON BAR SUPPORTED
BYRODS
A load is applied to a rigid bar that is symmetrically supported by three steel rods as
shown in Fig. 19. The cross-sectional areas of the rods are: rods A and C, 1.2 in^2 (7.74
cm^2 ); rod B, 1.0 in^2 (6.45 cm^2 ). Determine the maximum load that may be applied, (a) us-
ing elastic design with an allowable stress of 22,000 lb/in^2 (151,690.0 kPa); (b) using
plastic design with a load factor of 1.85.
Calculation Procedure:- Express the relationships among the tensile stresses in the rods
The symmetric disposition causes the bar to deflect vertically without rotating, thereby
elongating the three rods by the same amount. As the first method of solving this prob-
lem, assume that the load is gradually increased from zero to its allowable value.