Advanced Methods of Structural Analysis

180 6 Deflections of Elastic Structures

Fig. 6.21 (a) Actual state,
unit state forAand
corresponding bending
moment diagrams. (b)Actual
state, unit state forAand
corresponding bending
moment diagrams

Unit state

for qA

Actual state

A B

e = ql^2 /8

c = ql^2 /2

l/ 2

MP

(kNm)

l/ 4

q

Centroid C

M = 1

d = 1

f = 1 yc

M

a = 0

b = 1

W

qA

Actual state

ql 2 /8 ql

(^2) /2
l/ 2 l/ 4
MP
(kNm)
DA
W
A B
q
Centroid
Unit state
for DA
P = 1
1 ⋅l/ (^21) ⋅l
yc
M (m)
a
b
The next step is “multiplication” of two bending moment diagrams. The area of
square parabola according to TableA.1is
̋D
1
3

ql^2
2
l:
Centroid of this diagram is located on the distancel=4from fixed support. Cor-
responding ordinateycfrom diagramMN of unit state is 1. Multiplication procedure
is presented in Table6.5.
This table also contains computation of required displacement using the Simpson
rule (6.23). Ordinatesaandbare taken from the bending moment diagrams for
actual and unit states, respectively, at the left end of a beam (pointA); ordinatese
andfare taken at the middle of the beamAB, and ordinatescanddat the right end
(pointB).