TABLE 9.1
Boundary Conditions
for Beams
Case
Description
Boundary Conditions
1
Clamped
w0
0w
0
0x
2
Pinned
(hinged, simply
supported)
w0
0 2w
0
0x2
3
Free, with axial
force
p
4
Free, with massless
rigid constraint
and axial force
0 2w
0
0x2
0w
0 2w
0
a EI 2 b p
0
0x
0x
0x
p is tensile. Replace p with p
for a compressive force. For
no axial force, p 0.
Valid at either end of the beam.
0w
0
0x
2
0w
0w
0
a EI 2 b p
0
0x
0x
0x
Rigid constraint does not
permit rotation, but can
move unimpeded vertically.
p is tensile. Replace p with
p for a compressive force.
For no axial force, p 0.
Valid at either end of the beam.
0 2w
0
0x
2
0w
0
a EI 2 b so kw
0x
0x
k: spring constant (no
resistance to torsion)
so 1 at x 0
so 1 at x L
k → q; that is, w 0, Case 2
k 0; that is, (EIw) 0,
Case 3 if p 0
p
5
Free, with a
translation spring
6
Free, with a torsion
spring
7
Pinned, with a
torsion spring
8
Free, with torsion
and translation
springs
Remarks
0w
so kt
0x
0x2
0 2w
0
aEI 2 b 0
0x
0x
EI
0 2w
w0
0w
0 2w
EI 2 so kt
0x
0x
EI
0 2w
0x2
s¿o kt
0 2w
0
aEI 2 b so kw
0x
0x
kt: spring constant (no
resistance to vertical motion)
so 1 at x 0
so 1 at x L
kt → q; that is, w 0,
Case 4 if p 0
kt 0; that is, w 0,
Case 3 if p 0
kt: spring constant (no
resistance to vertical motion)
so 1 at x 0; so 1
at x L
kt → q; that is, w 0,
Case 1
kt 0; that is, w 0,
Case 2
0w
0x
k: spring constant (no
resistance to torsion)
kt: spring constant (no
resistance to vertical motion)
so 1 at x 0; so 1
at x L
(continued)
9.2 Governing Equations of Motion
TABLE 9.1
(continued)
Case Description
8
(continued)
9
Free, with mass
attached
10
Boundary Conditions
559
Remarks
so 1 at x 0; so 1
at x L
k → q; Case 7; k 0,
Case 6
kt 0; Case 5; k → q and
kt → q, Case 1
kt → q and k 0, Case 4
if p 0
EI
0 2w
so Jo
0 3w
0x
0x0t2
2
0
0 2w
0w
a EI 2 b s¿o Mo 2
0x
0x
0t
Free, with mass and
translation spring
attached
EI
2
0 2w
0x
2
so Jo
0 3w
0x0t
2
0
0 2w
0 2w
aEI 2 b s¿o Mo 2 s¿o kw
0x
0x
0t
Mo: attached mass
Jo: mass moment of inertia
of Mo
so 1 at x 0 so 1
at x L
so 1 at x 0 so 1
at x L
Mo 0 and Jo 0, Case 3
if p 0
Mo: attached mass
Jo: mass moment of inertia
of Mo
k: spring constant (no
resistance to torsion)
so 1 at x 0; so 1
at x L
so 1 at x 0 so 1
at x L
Upon taking the limits k1 씮 q and kt1 씮 q , these equations lead to the
boundary conditions given by Eqs. (9.42a) and (9.43a), respectively; that is,
the displacement is zero and the slope is zero. Similarly, if one were to use
Eqs. (9.47b) and considers the limits the translation stiffness k2 씮 q and the
torsion stiffness kt2 씮 q , then we arrive at the boundary conditions given by
Eqs. (9.44a) and (9.45a). Thus, we can think of a clamped end as a boundary
with infinite translation stiffness and infinite rotation stiffness.
If we consider the boundary conditions for a pinned end, which is the second entry of Table 9.1, this boundary is thought of as having infinite translation stiffness and zero torsion stiffness and zero rotary inertia. In the case of
the free end without an axial force—that is, a special case of the third entry of
Table 9.1—we see that this boundary condition is thought of as having zero
translation stiffness, zero torsion stiffness, zero translation inertia, and zero
rotary inertia. In practice, most boundary conditions lie between the two limiting cases; that is, between a free end and a fixed end.
In Section 9.3, we examine the free oscillations of beams. Before doing
so, we make several simplifying assumptions to reduce the algebra in the development. First, we assume that the beam is homogeneous and that it has a
uniform cross-section along its length; that is,
EI1x2 EI,
r1x2 r,
and
A1x2 A
(9.48)