Advanced Methods of Structural Analysis

Appendix 575

A. Foci points:

Each span of a beam contains two foci points. They are left and right points. If
loadedspans are located to therightof the spanln, then the bending moment dia-
gram in all left spans is passing throughleftfoci points only. These foci points are
indicatedFnL;FnL 1.
If loaded spans are located to theleftof the spanln, then the bending moment
diagram in all right spans is passing throughrightfoci points only, which indicated
asFnR;FnRC 1.
The left and right foci quotients connect consecutive support moments as follows:

knL 1 D

Mn 1

Mn 2

;knLD

Mn

Mn 1

knRC 1 D

Mn

MnC 1

;kRnD

Mn 1

Mn

n

* *

FnL FnR

* *

Loaded span

n− (^2) ln
n− 1
ln− 1 Mn−^1
Mn− (^2) FnL− 1 F
nR− 1
Mn
Elastic curve

Loaded span
FnL
FnR
ln
n− 1
ln+ 1
ln+ 1
Mn+ 1
Mn− 1
FnL+ 1 FnR+ 1
(^01)
M 1

• 0
  1
  M 1
  F 1 L F 1 L
  M 0


• 
  

  l 1 l 1
  c n
  c 1 2

  
  


• 
  

  ---

  
  

  ln
  n− 2
  n− 1
  n+ 1 n+^2
  ln− 1 ln− 1 ln+ 1 ln+ 2
  Mn− 1 Mn
  FnL− 2 FLn− 1 FnR+ 1 FnL+ 2
  Fig. A.1 (a) Explanation of the left foci points. (b) Explanation of the right foci points. (c)Fo-
  cus relationships for pinned and fixed supports. (d) Continuous beam. Notation, bending moment
  diagram in the given structure due to applied load, left and right foci points