Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

446
STATICALLy INdETERMINATE
bEAMS
M A
w
A x A
B
A y B y
A x
P
(a) Actual beam with loads
and reactions
M A
A y
A
w
(b) Released beam if B y is
chosen as the redundant
P
w
A x
A
B
A y
(c) Released beam if M A is
chosen as the redundant
FIGURE 11.1 Propped
cantilever beam.
P
B
vertical directions, respectively, and a restraint M A against rotation. The roller support prevents
translation in the vertical direction (B y ). Consequently, the propped cantilever has
four unknown reactions. Three equilibrium equations can be developed for the beam (ΣF x = 0,
ΣF y = 0, and ΣM = 0), but since there are more unknown reactions than there are equilibrium
equations, the propped cantilever is classified as statically indeterminate. The
number of reactions in excess of the number of equilibrium equations is termed the degree of
static indeterminacy. Thus, the propped cantilever is said to be statically indeterminate to
the first degree. The excess reactions are called redundant reactions or simply redundants
because they are not essential to maintaining the equilibrium of the beam.
The general approach used to solve statically indeterminate beams involves selecting
redundant reactions and developing an equation pertinent to each redundant on the basis of
the deformed configuration of the loaded beam. To develop these geometric equations, redundant
reactions are selected and removed from the beam. The beam that remains is called
the released beam. The released beam must be stable (i.e., capable of supporting the
loads) and statically determinate so that its reactions can be determined by equilibrium
considerations. The effect of the redundant reactions is addressed separately, through
knowledge about the deflections or rotations that must occur at the redundant support. For
instance, we can know with certainty that the beam deflection at B must be zero, since the
redundant support B y prevents movement either up or down at this location.
As mentioned in the previous paragraph, the released beam must be stable and statically
determinate. For example, the roller reaction B y could be removed from the propped cantilever
beam (Figure 11.1b), leaving a cantilever beam that is still capable of supporting the applied
loads. In other words, the cantilever beam is stable. Alternatively, the moment reaction M A
could be removed from the propped cantilever (Figure 11.1c), leaving a simply supported
beam with a pin support at A and a roller support at B. This released beam is also stable.
A special case arises if all of the loads act transverse to the longitudinal axis of the beam.
The propped cantilever shown in Figure 11.2 is subjected to vertical (transverse) loads only. In
this case, the equilibrium equation ΣF x = A x = 0 is trivial, so the horizontal reaction at A vanishes,
leaving only three unknown reactions: A y , B y , and M A . Even so, this beam is still statically
indeterminate to the first degree, because only two equilibrium equations are available.
Another type of statically indeterminate beam is called a fixed-end beam or a fixedfixed
beam (Figure 11.3). The fixed connections at A and B each provide three reactions.
Since there are only three equilibrium equations, this beam is statically indeterminate to the
third degree. In the special case of transverse loads only (Figure 11.4), the fixed-end beam
has four nonzero reactions but only two available equilibrium equations. Therefore, the
fixed-end beam in Figure 11.4 is statically indeterminate to the second degree.
The beam shown in Figure 11.5a is called a continuous beam because it has more
than one span and the beam is uninterrupted over the interior support. If only transverse
loads act on the beam, it is statically indeterminate to the first degree. This beam could be
released in two ways. In Figure 11.5b, the interior roller support at B is removed so that the
released beam is simply supported at A and C, a stable configuration. In Figure 11.5c, the
exterior support at C is removed. This released beam is also simply supported; however, it
now has an overhang (from B to C). Nevertheless, the beam’s configuration is stable.
M A
w
P
M A
P
w
M B
M A
P1
P2
M B
A
A x
A y B y
B
A
A x
A y B y
B
B x
A
A x
A y B y
B
B x
FIGURE 11.2 Propped cantilever
subjected to transverse loads only.
FIGURE 11.3 Fixed-end beam with
load and reactions.
FIGURE 11.4 Fixed-end beam with
transverse loads only.