Advanced Methods of Structural Analysis

6.3 Maxwell–Mohr Method 163

(b)The vertical displacement atAmay be calculated by formula

yAD

1

EI

Zl

0

Mp.x/MNdx; (c)

where expression for bending momentMp.x/in the actual state remains without
change. In order to construct the unit state, it is necessary to apply unit concentrated
forcePD 1 at the point where it is required to determine displacement (Fig.6.10b).
For unit state, the bending moment isMD 1 x. The formula (c) for vertical
displacement becomes

yAD

1

EI

Zl

0

.

qx^2

2

/. 1 x/dxD

ql^4

8 EI

: (d)

The positive sign means that adopted unit load make positive work on the real dis-
placement, or other words, actual displacement coincides with assumed one.

Example 6.8.Determine the vertical displacement of joint 6 of symmetrical truss
showninFig.6.11. Axial rigidity for diagonal and vertical elements isEAand for
lower and top chords is 2EA.

h = 3m

P

1

P

d = 4m

3795

4 68 10

2

RA = P RB = P

a

Actual

state

y 6

2 EA

2 EA

EA EA

P = 1

_

RA = 0.5

_

RB = 0.5

Unit

state

1 35

(^246)
(^79)
8
10
sin ̨D3=5
cos ̨D4=5
Fig. 6.11Design diagram of the truss (actual state) and unit state
Solution.All elements of the given structure are subjected to axial loads only, so
for required displacement the following formula should be applied:
y 6 D
X 1
EA
NpNNkl; (a)
whereNpandNkare internal forces in actual and unit state, respectively.