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Chapter 6: Influence Lines for Statically Determinate Structures CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Distributed LiveLive Load Distributed Load Distributed Live Load w Concentrated Live Load Concentrated Live Load Concentrated Live Load Why Influence Lines? P Dead Load A B M V w, P Analysis D C V, M Design Note: loads can vary- LIVE LOADS Vmax and Mmax under dead & live loads? Can my bridge survive? What is Influence Line ? 1 2 6 (1 6 8 ) 8 5 (1 2 7 ) Force 1 (4 3 ) 4 2 (8 4 ) 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 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 M B 0 A y (10 ) (10 x )( 1) 0 Ay 1 1 x 10 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 F1 r1 F i 0 r3 F3 i r2 F2 r Virtual deformation A structure in equilibrium: Fi i Virtual work: F4 0 r4 imaginary deformation Resultant force=0 W Fi r 0 i W Fi ri 0 i Virtual work principle for a rigid-body system Müller Breslau Principle Influence line = properly disturbed shape x 1 By Ay 1 1 1 Ay f(x) Ay By Virtual work δW 0 Influence line of Ay W A y 1 1 f(x) 0 A y f(x) Müller Breslau Principle Influence line = properly disturbed shape 1 x Ay By 1 1 1 f(x) Ay Virtual work δW 0 By By Influence line of By W B y 1 1 f(x) 0 B y f(x) Müller Breslau Principle Influence line = properly disturbed shape Include ONLY the reaction/force in the virtual work 1 x V Ay V By 1 M 1 D2 q1 q2 D1 Ay f(x) By M V W V D 1 D 2 M q 2 q1 1 f(x) 0 V f(x) 1 0 Müller Breslau Principle Influence line = properly disturbed shape Include ONLY the reaction/force in the virtual work 1 x M Ay q1 M V By 1 1 V D q2 f(x) By Ay W M q1 q 2 1 f(x) 0 M f(x) 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 P 1 1 SB A w l dx wl MB B D C P SB y P MB a b M B y w l dx b b M B a y w l dx w l a ydx w const l Distributed LiveLive Load Distributed Load Distributed Live Load Concentrated Live Load Concentrated Live Load Concentrated Live Load Distributed Loads Dead Load A B C D 1 A SB MB Concentrated Live Load Application of Influence Lines B P Distributed Live Load wl Given dead load and live loads Find maximum forces wd C D Application of Influence Lines • Max shear force at C? Application of Influence Lines • Max shear force at C? Case 1 : (V C ) 1 4 . 5 ( 0 . 75 ) 18 ( 0 . 625 ) 18 ( 0 . 5 ) 23 . 63 kN Case 2 : (V C ) 2 4 . 5 ( 0 . 125 ) 18 ( 0 . 75 ) 18 ( 0 . 625 ) 24 . 19 kN Case 3 : (V C ) 3 4 . 5 ( 0 ) 18 ( 0 . 125 ) 18 ( 0 . 75 ) 11 . 25 kN 6. Influence Lines • What is influence line? • Müller Breslau Principle • What is the use of influence line?