Influence Line - Lecture Notes 1

Influence Lines for Statically Determinate Structures Objectives: Influence lines have important application for the design of structures that resist large live loads. The objectives are (i) Understand the moving load effect in simpler term (ii) Study various definitions of influence line (iii) Introduce to simple procedures for construction of influence lines for statically determinate structure. Topics Influence Lines for Statically Determinate Beams Computing Forces from Influence Lines Influence Lines for Trusses Solved examples and tutorials Can my bridge survive? What is Influence Line ? 126 (168) 85 (127) 1 (43) Force 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 P C A x RA  B Px L P RB P B A B A 8 5 P R B L=10 RA RA =0.5P P (L x) L RA=0.2P L=10 RB=0.5P 0.5P RA x Plot for RA 0.5P Plot for RB L=10 P RB RB=0.8P x RA RB 0 1.0P 0 2 0.8P 0.2P 4 0.6P 0.4P 5 0.5P 0.5P 6 0.4P 0.6P 8 0.2P 0.8P 10 0 1.0P 6-1 Influence Lines If a structure is subjected to a moving load, the variation of shear & bending moment is best described using the influence line Influence lines play an important part in the design of bridges, industrial cranes, conveyors and other structures where loads move across their span. So what is an influence line? (see previous slide) An influence line represents the variation of either the reaction, shear, moment or deflection at a specific point in a member as a concentrated force moves over the member. Once the influence line is drawn, the location of the live load which will cause the greatest influence on the structure can be found very quickly. Influence lines represent the effect of a moving load only at a specified point on a member, whereas shear (SFD) and bending moment diagrams (BMD) represent the effect of fixed loads at all points along the member. Procedure to construct the influence line (three methods) 1) Tabulate Values 2) Influence-Line equations 3) Virtual work: Muller-Breslau principle (qualitative influence line) Construction of influence line by using equn & tabulated values Example 1: Construct the influence line for the vertical reaction at A of the beam in Fig.1(a) x 10 m Ay Fig 1 (a) Tabulate Values: A unit load is placed on the beam at each selected point x and the value of Ay is calculated by summing moments about B. x Ay 1 By 10 m (b) Text book example 6-1 1 10 m By (c) Influence-Line Equation: When the unit load is placed a variable distance x from A, the reaction Ay as a function of x can be determined from M B 0; Ay (10)  x)(1) 0 (10  1 Ay 1  x 10 This line is plotted in Fig. (c) EXAMPLE 2: Construct the influence line for the vertical reaction at B of the beam in Fig. 2a. x 1 5m Ay Fig. 2a 5m By Solution Tabulate Values: A unit load is placed on the beam at each selected point x and the value of By is calculated by summing moments about B. A plot of the values yields the influence line in Fig. c. Text book example 6-2 x 1 5m 5m By (c) Influence-Line Equation. Applying the moment equation about A, M A 0; By (5) 1( x)  0 1 By  x 5 This line is plotted in Fig. (c) EXAMPLE 3: Construct the influence line for the shear at point Text book example 6-3 C of the beam in Fig. 3a. Tabulate Values: The unit load must be placed just to the left x=2.5- and just to the right x=2.5+ of point C since the shear is discontinuous at C, Figs. 6–3b and 6–3c. A plot of the values in Fig. 6– 3d yields the influence line for the shear at C, Fig. 6–3e. Fig. 3a Solution 0.25 (b) (c) A C B (e ) Influence-Line Equation. Here two equations have to be determined since there are two segments for the influence line due to the discontinuity of shear at C, 1 Vc   x 10 1 Vc 1  x 10 These lines are plotted in Fig. (e) EXAMPLE 4: Construct the influence line for the shear at point C of the beam in Fig. a. Text book example 6-4 x 1 B C A 4m 4m 4m Solution x Tabulate Values: Using statics and the method of sections, get VC for different position of unit load. Equation EXAMPLE 5: Construct the influence line for the moment at point C of the beam in Fig. a. x 1 B C A 5m 5m (a) Solution Tabulate Values: A B C B Free-body of BC Text book example 6-5 Tabulated values (c) Influence-Line Equations: MC MC 1 x M C 0 C A 1 x)  1 (5  (1  x)5 0 10 1  x for 0 x 5m 2 5m M C 0 B 5m 1.25 1.25 1 MC  (1  x)5 0 10 1 MC  5  x for 5 m x 10m 2 EXAMPLE 6: Construct the influence line for the momentr at point C of the beam in Fig. a. x 1 B C A 4m 4m 4m Solution x Tabulate Values: Using statics and the method of sections, get MC for different position of unit load. Equation Text book example 6-6 6-3 Qualitative Influence Lines Muller-Breslau Principle states that the influence line for a function (reaction, shear, moment etc..) is to the same scale as the deflected shape of the beam when the beam is acted upon by the function. - A quicker way of constructing influence lines for beams is to apply the Müller Breslau Principle, which is derived from the principle of virtual work. For example to obtain the influence line for the support reaction at “A” for the beam shown, remove the support corresponding to the reaction and apply a force in the +ve direction that will cause a unit displacement in the direction of Ay Notice that the deflected shape is linear (i.e. the beam rotates as a rigid body without any curvature). This is true only for statically-determinate beams. 1 1 If the shape of the influence line for shear at C is to be determined, Fig 6.13(a), the connection at C may be symbolized by a roller guide as shown in Fig 6.13(b). Applying a +ve shear force Vc to the beam at C & allowing the beam to deflect to the dashed position, the influence line shape as shown in Fig 6.13(c) Fig. 6-13 If the shape of influence line for the moment at C Fig 6.14(a) is to be determined, an internal hinge or pin is placed at C  Applying +ve moment Mc to the beam, the beam deflects to the dashed line which is the influence line as shown in Fig 6.14(c) Fig. 6-14 The proof of the Muller-Breslau Principal can be established using the principle of virtual work Principle of Virtual Work States that for an equilibrium system, the work done by all forces upon a set of virtual displacement is zero. Work = a linear disp.  force in the direction of disp. Or work = rotational disp.  moment if the direction of the disp. For the FBD shown, the only forces having a corresponding virtual displacement are Ay and the unit load, thus: y/ .1  0; 1Ay    y/ Ay  FBDs of a beam and a virtually-displaced beam x 1 A C By Ay x 1 A C y=1 y/ Ay B B By This indicates that the influence line of Ay is numerically equal to the virtual displacement of the beam, when the virtual displacement is constructed with a unit displacement at Ay and no displacement at any forces except the unit load. Müller Breslau Principle The step-by-step process of applying the Müller Breslau Principle can be summarized as follows: 1. Expose the quantity of interest by a cut (or remove a support). 2. Impose a virtual displacement such that: (a) at the cut there is a unit displacement (or rotation). (b) the quantity of interest produces a positive work. (c) no other internal force produces any work. 3. The resulting displacement shape is the desired influence line. Notes 1. The Müller Breslau Principle provides a quick method for establishing the shape of the influence line. The generated shape is referred to as a qualitative influence line. Once this is known, the ordinates at the peaks can be determined by using the basic method introduced earlier. 2. By simply knowing the general shape of the influence line, it is possible to locate the live load on the beam and then determine the maximum value of the function using statics Example:ShearforceandMomentatC If the beam is sectioned at C, the beam undergoes a virtual disp y at this point, Fig 6.15(c), then only the internal shear at C and the unit load do work The virtual work eqn is: Vcy 1y ' 0 If y  1, then  Vc y ' Fig. 6-15 Assume a hinge or pin is introduced into the beam at point C, Fig 6.15(d). If a virtual rotation is introduced at the pin, virtual work will be done only by the internal moment & unit load Mc 1 y' 0 If   1, then  M c  y ' Fig. 6-15 This indicates that the deflected beam has the same shape as the influence line for the internal moment at point C