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

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

3. Can my bridge survive?

4. What is Influence Line ? Force

5. Influence Lines Measurement

6. KSM Influence Lines Measurement

7. KSM Influence Lines Measurement

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

9. 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

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

11. Point-by-point calculation • Construct the influence line for Ay

12. Influence-line equation Linear function of x

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

14. 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

15. Müller Breslau Principle

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

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

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

19. Müller Breslau Principle 1 Ay By Ay By 1 V V M D2 q2 q1 D1 M f(x) = = 1 0 V Influence line = properly disturbed shape Include ONLY the reaction/force in the virtual work 1 x

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

21. 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.

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

23. Influence Lines for Trusses

24. Example 6.15 • Draw the influence line for member force GB

25. Example 6.15 • Draw the influence line for the member force GB

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

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

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

29. Application of Influence Lines • Max shear force at C?

30. Application of Influence Lines • Max shear force at C?

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