Advanced Methods of Structural Analysis

354 10 Influence Lines Method

l

l l

a P= 1 P= 1

i=EI/l

b

i i

Z 1

1

r 11

3 i^4 i

3 i

Z 1 =1

3 i

3 i

2 i

4 i

k

Mk = −1.2i

Qk = −^3 li M^1

r 11

×

Mk

c

Pu^2 ul

r 1 P

P

ul ul

P in right span

Puu^2 l

r 1 P = −u( 1 −u)^2 l

P in left span r 1 P

P

ul ul

r 1 P=ul 2 ( 1 −u^2 )

Pu 2 ( 1 −u (^2) )⋅l
d
016234578910
Z 1
0.00960.0168
0.01280.01440.00960.0032
0.01920.0144 Inf. line Z 1
Factor l^2 /EI

• −
  e
  Fig. 10.15 (a,b) Design diagram and primary system. (c) Unit state and corresponding bending
  moment diagram. (d) Calculation of free termr1P.(e) Influence line for primary unknownZ 1
  (TableA.3), instead of parameterwe must write the parameteru. Therefore, if
  loadPD 1 is located in the left span, then the free term of canonical equation
  isr1P D u 2 l
  
  1 u^2
  
  .IfloadP D 1 is located in the right span, then we can
  directly apply formula from TableA.4,sor1PDu^2 lDu.1u/^2 l.
  3.Influence line for primary unknown.Having expressions forr1P in terms of
  positionPandr 11 D10i, the expressions for primary unknownZ 1 may be
  presented as follows:
  PD 1 in the left span
  IL.Z 1 /D
  1
  10i
  ul
  2
  
  1 u^2
  
  D
  u
  20
  
  1 u^2
  l^2
  EI
  :