9.Influence - Influence Lines Theory and Design of...

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Theory and Design of Structures I Influence Lines Influence Lines Moving Loads In static, constant and stationary loads, the reactions, stresses and deformations at a particular section are constant However, the load effects become variable functions of the position of the load when the loads are moving, even if the dynamic effects are ignored Influence Lines Examples of moving loads: locomotives rolling on a railway bridge motor vehicles speeding over flyovers occupants moving in a building Two structural problems arise: position of the live load that will cause the maximum effect at a certain section magnitude of the maximum effect Influence Lines An influence line is a graph or chart showing the value of some direct linear function (reaction, shear, bending moment, axial force, etc) at a certain point in the structure due to a unit concentrated load moving across the span It illustrates the variation of the effect Q (reaction, stress) at a fixed point i due to a moving unit cause (unit force or moment) acting at a moving position j Use of Influence Lines Use of Influence Lines It provides a tool for determining the maximum load effect. There are 3 different cases in determination of the value of the effect Q at a fixed point i: Single concentrated moving load System of concentrated moving loads Uniformly distributed moving load P = 1 i j z(j) j ) ( j z P Q i = i P = j j j i z P Q i P 3 P 1 P 2 z 1 z 2 z 3 = = A p pzdx Q i i p z A d
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P = 1 i j z(j) j ) ( j z P Q i = i P Single concentrated moving load Influence line due to unit load System of concentrated moving loads P = 1 i j = j j j i z P Q i P 3 P 1 P 2 z 1 z 2 z 3 Uniformly distributed moving load P = 1 i j = = A p pzdx Q i i p z A d Influence Lines for Beams Consider a simply supported beam with overhang as shown A B D C 4m 4m 16m Moving unit load R B 1.25 A B D C 4m 4m 16m x P = 1 I.L. for reaction at B, R B Imagine a unit load P starts at C and moves towards B and A so that distance x of the load from C is varying. Taking moment about C for the whole beam, Px L R B = 16 x R B = 20 0 x I.L. for R B R B
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R B 1.25 A B D C 4m 4m 16m x P = 1 I.L. for reaction at B, R B When P is at C, When P is at B, When P is at A, 0 , 0 = = B R x 1 , 16 = = B R x 25 . 1 , 20 = = B R x I.L. for R B R B I.L. for reaction at B, R B I.L. for reaction at C, R C Imagine a unit load P moves from A to C so that x is the distance of P to the right of B. Taking moment about B for the whole beam, R C (16) = P x R C = x /16 for -4 x 16 1.0 R C -0.25 A B D C 4m 4m 16m x P = 1 I.L. for R C R C I.L. for reaction at C, R C To draw I.L. for shear at section immediately to the
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