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

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.

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

Influence lines Both Qualitative Influence Lines Quantitative

Quantitative Influence Lines

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.

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

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

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

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

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.

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

Influence Line Ordinates Unit Load at Influence Line Ordinates By Mc A 1.5 -1.5 B 1 0 C 0.519 1.56 D 0.148 0.44 E 0 0

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

Example 2 1k

Qualitative Influence Lines

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”

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

Example 1: The same beam of our Lecture Hall D B 1KN ILD for Ey

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

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

Further Reading – Web Resources www.ecf.utoronto.ca/apsc/courses/civ214/Lectures/Influence_lines.html http://en.wikipedia.org/wiki/Category:Structural_analysis www.childs-ceng.demon.co.uk/tutorial/tuIL.html http://www.public.iastate.edu/~fanous/ce332/influence/twospanbeam.html

The End Questions