bridges In recent years the MEXE method has been the subject of some criticism in particular with respect to determining the carrying capacity of short span bridges Havey 2007 McKibbins et al bri ID: 832126
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Over the years there have several attemp
Over the years there have several attempts to develop assessment methods for masonry arch bridges. In recent years the MEXE method has been the subject of some criticism in particular with respect to determining the carrying capacity of short span bridges (Havey 2007, McKibbins et albridges. New equations are introduced and comparisons have been made with the results using Pippard’s original equations (Pippard 1948). It a
lso attempts to identify other limits of
lso attempts to identify other limits of Pippard’s elastic method. 2 PIPPARD’S ELASTIC METHOD FOR SHORT SPAN ARCH BRIDGES Pippard used Castigliano's theorems that the parrespect to a force, is equal to the displacement in the direction of the force. He (Pippard 1948) treated the ring as a two-pin parabolic arch with a secant variation of sec, as shown in Jinyan Wang, Clive Melbourne and Adrienn Tomor
491 Figure 1 : Pippa
491 Figure 1 : Pippard’s two-pinned parabolic arch (Additionally, Pippard ignored the axial thrust and shearing force terms in the strain energy equation. Hence the strain energy was assumed to be totally dependent upon the flexural response of the arch, i.e. (1) is modulus of elasticity and d is increment of length along the arch ring. Thus th
e value of horizontal reaction at the a
e value of horizontal reaction at the abutment is given by the solution of the equation (2) Total bending moment at is given by where is the statically determinate bending moment, thereforeyHM. Substitute the relationships into Eq.(2) gives (3) Pippard considered a secant variation of the second moment of area, that is,
is given by BABAsBABAsBABAsdxyydxMEId
is given by BABAsBABAsBABAsdxyydxMEIdxyEIdxyMEIdsyEIdsyMH20202secsecsecsec (4) is increment of length along span L However, for a short span arch with relatively a thicker ring, the axial thrust term in the strain energy U should be considered, i.e. (5) is the cross section area of the arch ring at any point at any point is given by cosHN,
(6) S
(6) Substituted Eq.(6) into Eq.(5) gives WJinyan Wang, Clive Melbourne and Adrienn Tomor 493 Table 1 : Comparison of live load effects Pippard’s equations New equations BABASEIdsyEIdsyMH2 BABABASEAdsEIdsyEIdsyMH22cos the crown due to a unit live load at the crown aLHL12825 LTLHH11 Bending moment at the crown due to a unit live load at the crown LML1287
LTLMM17321 Stress at the crown extrad
LTLMM17321 Stress at the crown extrados due to a unit live load at the crown (effective width of 2dahdLhdMhdHLL4225256322 7321422525611322dahdLhdMhdHTLTL is arch central rise, is ring thickness at the crown, is fill depth at the crown, L is is density of the fill and masonry (assumed to be the same). Table 2 : Comparison of dead load effects Pippard’s equations New equations Horizontal thrust at the crown of a unit
width bridge 21422adhaLHD 1121422adhaL
width bridge 21422adhaLHD 1121422adhaLHTD Bending moment at the crown of a unit width bridge 3362aLMD 1421713362adhaLMTD Stress at the crown extrados due to the dead load daadhdLdMdHDD2842112622 adhdaadhdLdMdHTDTD4271284211112622 In accordance with Pippard, for a point load at the crown the compressive stress under the combined dead and live load together should not exceed the maximum permitted value of the compressive st