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Unformatted text preview: 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 P2 z 1 z 2 z 3 ∫ ⋅ = = A p pzdx Q i i p z A d 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 ≤ ≤ x I.L. for R B R B 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, , = = 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 C0.25 A B D C 4m 4m 16m x P = 1 I.L. for R C R C...
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This note was uploaded on 12/01/2010 for the course CIVL 2007 taught by Professor Profchai during the Spring '10 term at HKU.
 Spring '10
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