CIVL3310 STRUCTURAL ANALYSIS Chapter 6: Influence Lines for Statically Determinate Structures CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang

Why Influence Lines? Vmax and Mmax under dead & live loads? Concentrated Live Load Concentrated Live Load Concentrated Live Load w P Distributed Live Load Distributed Live Load Distributed Live Load Dead Load A B C D V M Analysis w, P V, M Design Note: loads can vary- LIVE LOADS Vmax and Mmax under dead & live loads?

Can my bridge survive?

What is Influence Line ? Force

Influence Lines Measurement

KSM Influence Lines Measurement

KSM Influence Lines Measurement

Influence Lines Shear force and moment diagrams Fixed loads V M x V and M at different locations of the beam x Influence Lines Load moves along the beam V M x x V and M at a fixed location

Influence Lines Influence line: A graph of a response function (such as reactions or internal forces) of a structure as a function of the position of a downward unit load moving across the structure Response Location of downward unit load

Constructing Influence Lines Point-by-point calculation Influence-line equation Graphical approach: Müller Breslau Principle

Point-by-point calculation Construct the influence line for Ay

Influence-line equation Linear function of x

Influence Lines All statically determinate structures have influence lines that consist of straight line segments More examples!

Müller Breslau Principle Influence line for any action (reaction, internal shear/moment) in a structure is equal to the deflection curve when we remove the action and replace it with a corresponding unit displacement or rotation Influence line = properly disturbed shape

Müller Breslau Principle

Virtual work principle for a rigid-body system Virtual deformation A structure in equilibrium: imaginary deformation Resultant force=0 Virtual work: Virtual work principle for a rigid-body system

Müller Breslau Principle Influence line = properly disturbed shape x 1 Ay By 1 Ay 1 Ay 1 f(x) By Influence line of Ay Virtual work

Müller Breslau Principle Influence line = properly disturbed shape x 1 Ay By 1 By 1 By Influence line of By 1 f(x) Ay Virtual work

Müller Breslau Principle Influence line = properly disturbed shape Include ONLY the reaction/force in the virtual work x 1 V Ay By V M f(x) 1 D2 1 q2 q1 Ay D1 By M V = 1 =

Müller Breslau Principle Influence line = properly disturbed shape Include ONLY the reaction/force in the virtual work x 1 M Ay By V M f(x) 1 D 1 q1 q2 Ay By = 1

Müller Breslau Principle Influence line = properly disturbed shape Deflect the structure such that only the force which influence line that you are looking for and the downward unit force contribute to the virtual work due to the imaginary deflection. All other forces that act on the virtually deflected structure should not contribute to the virtual work.

Influence Lines for Floor Girders Draw the influence line for the shear in panel CD of the floor girder

Influence Lines for Trusses

Example 6.15 Draw the influence line for member force GB

Example 6.15 Draw the influence line for the member force GB

Application of Influence Lines 1 1 wl SB MB A B C D a b SB MB y

Distributed Loads Concentrated Concentrated Concentrated Live Load Distributed Live Load Distributed Live Load Distributed Live Load Dead Load A B C D

Application of Influence Lines Concentrated Live Load P Given dead load and live loads Find maximum forces Distributed Live Load wl 1 wd A B C D SB MB

Application of Influence Lines Max shear force at C?

Application of Influence Lines Max shear force at C?

6. Influence Lines What is influence line? Müller Breslau Principle What is the use of influence line?