Civil Engineering Reference
In-Depth Information
represented as short beams and, as such, the beam bending deflection can be
neglected under transverse loads. Errors caused by this approximation can be
calibrated using FEM.
Many low-rise rigidly framed structures are designed using repeating fixed
beam and column configurations. For these structures, Iskander et al. (2012a)
stipulate that the lateral deflection at any elevation within a low rise rigid frame
is dependent on ten variables: (1) modulus of elasticity,
E
; (2) Poisson's ratio,
;
(3) total acting force,
W
; (4) moment of inertia of the individual columns,
I
c
; (5)
moment of inertia of the individual beams,
I
b
; (6) the height of the columns,
l
c
;
(7) the length of the beams,
l
b
; (8) the total number of stories,
s
; (9) the total
number of bays,
b
; and (10) the story from the top where the deflection is
desired,
i.
ν
3.3.1 Derivation of Equations for Lateral Deflection,
δ
s
The slope of the deflection curve of a beam due to shear force,
d
δ
S
/dz
, is approx-
imately equal to the shear strain,
, of any beam (Gere & Timoshenko 1984). As-
suming that the material is linear homogenous and isotropic, and applying
Hooke's law, the shear strain,
γ
γ
, is expressed as follows:
d
δ
s
Q
GA
o
dz
= γ =
(3.1)
where,
Q
is the total applied shear force, and
A
o
is the equivalent area over which
the shear stress is acting. The lateral deflection can be obtained by integration:
Q
GA
0
z
=
H
d
δ
=
dz
δ
s
=
(3.2)
z
=0
where,
H
is the total height of the frame, and
z
is the distance from the top of
the frame to the point where the lateral deflection is to be determined. At the bot-
tom of the frame,
z=H
and at the top of the frame
z=0
(Fig. 3.4). The total shear
force,
Q
, can be expressed as a function of the shear force distribution along the
height of frame, as follows:
()
⋅
d
Q
=
qz
(3.3)
An equation for the shear force as a function of the distance,
z
, from the top of
the frame, can be obtained by evaluating the expression for the applied load distri-
bution shown in Fig. 3.1. For example, a hydrostatic pressure, with a magnitude
