Advanced Methods of Structural Analysis

146 6 Deflections of Elastic Structures

solution of this problem. At present, methods for computation of the displacements
are developed with sufficient completeness and commonness for engineering pur-
poses and are brought to elegant simplicity and perfection.
The deformed shape of a bend structureis defined by transversal displacements
y.x/of every points of a structural member. The slope of the deflection curve is
given by.x/D dy=dx Dy^0 .x/. Deflected shapes of some structures are pre-
sented in Fig.6.1. In all cases, elastic curves (EC) reflect the deformable shape of
the neutral line of a member; the EC are shown by dotted lines in exaggerated scale.

b P

A B

C

DC

qA qB

a

A

B

P

q qB DB

A=^0

x

Tangent at B

y

y(x)

q (x)

c

A B

P

C D

qB

DC DD

*Inflection

point

d

A B

P

C D

DC DD

eAHB

D

f AB

D

Fig. 6.1 (a–d) Deflected shapes of some structures. (e,f) Deflected shape of beams caused by the
settlementof supportB

A cantilever beam with loadPat the free end is presented in Fig.6.1a. All points
of the neutral line have some vertical displacementsy.x/. EquationyDy.x/is the
EC equation of a beam. Each section of a beam has not only a transversal (vertical
in this case) displacement, but an angular displacement.x/as well. Maximum
vertical displacementBoccurs atB; maximum slopeBalso happens at the same
point. At the fixed supportA, both linear and angular displacementsAandAare
zero.
The simply supported beam with overhang is subjected to vertical loadPas
shown in Fig.6.1b:The vertical displacements at supportsAandBare zero. The
angles of rotationAandBare maximum, but have different directions. Since
overhangBCdoes not have external loads, the elastic curve along the overhang
presents the straight line, i.e., the slope of the elastic curvewithin this portion is
constant. The angles of rotation of sections, which are located infinitely close to the
left and right of supportBare equal.