1. PROCLAIM !!!
IN THE NAME OF THY
LORD
WHO CREATETH

2. Continuous and Fixed Ended Beams
Theory of Structures
Influence Lines
For
Continuous and Fixed Ended Beams
By Fawad Ahmed Najam 05-CE-31

3. Influence Lines
An influence line is a graph of a response function of a structure as a function of the position of a downward unit load moving across the structure.
Response function can be reactions, shears and moments.
For indeterminate structures, Influence lines are generally curved.
Any of the methods of analysis of indeterminate structures can be used for computing the ordinates of influence lines.

4. Importance of Influence Lines
Their primary use is in calculating the maximum response (e.g. bending moment, shear, or deflection) at a given point in the structure due to moving live loads due to the passage of heavy trucks, trains, and other similar causes.
Example of bridge collapse in Karachi

5. Influence lines
Both
Qualitative
Influence Lines
Quantitative

6. Quantitative Influence Lines

7. Procedure for analysis
Compute the displacement of successive points along the beam when the beam is subjected to unit load at a point of which influence line is to be drawn.
Divide each value of displacement by the displacement determined at the point where the unit load acts.
The resulting values are the ordinates of the influence line.

8. Example 1 x
1KN B C D E A By 3m 3m 3m 3m

9. x
1KN B C D E A
fBX
- fBB B C D E A
1KN

10. x
1KN B C D E A By 3m 3m 3m 3m
fDB
1KN B C D A
fCB E
fBB
fAB

11. Solution Compatibility Equation: By = fBX / fBB fBX fBB (By) - Where,
= 0 -
By = fBX / fBB
Where,
f BB denotes the downward deflection at B due to unit load

12. Solution
We will compute the deflections at points A through E using the beam deflection formulas given inside the front cover of various books e.g. “Structural Analysis” by Aslam Kassimali. (or use conjugate beam method)
Dividing these deflections by deflection at B (fBB) one by one, we will get the ordinates of influence line diagram for By.

13. Maxwell’s law of reciprocal deflections
“Deflection at any point, say A, due to a unit load at B must be equal to the deflection at B due to unit load at A”
fBA
fAB
-364.5/EI
fBB
-243/EI
fBC
fCB
-126/EI
fBD
fDB
-36/EI
fBE
fEB

14. Influence Line Ordinates
Unit Load at
Influence Line Ordinates
By Mc A B C D E

15. 1KN B C D E A By
1.5 1
0.519 By
0.148
1.56
0.44 Mc
1.5

16. Example 2 1k

20. Qualitative Influence Lines

21. Müller - Breslau's Principle
Statement:
"The Influence line for a force (or moment) response function is given by the deflected shape of the released structure obtained by removing the restraint corresponding to the response function from the original structure and by giving the released structure a unit displacement (or rotation) at the location and in the direction of response function, so that only the response function and the unit load perform external load”

22. Example 1: The same beam of our Lecture Hall
1KN x A E C D B Ey
1KN x A Ey

23. Example 1: The same beam of our Lecture HallD B
1KN
ILD for Ey

24. Example 2
1KN A B C D E F AY DY FY VC Mc

25. Example 3
1KN D B A C
9ft
9ft
18ft MD

26. Further Reading – Web Resources

27. The End
Questions