Shear Lag 322021 Prof Carlos Montestruque 1 Shear

Shear Lag 3/2/2021 Prof. Carlos Montestruque 1

Shear Lag SHEAR LAG Thin sheet structures under loading conditions that produce characteristically large and non-uniform axial (stringer) stress. More pronounced in shells of shallow section than in shells in deep section. Much more important in wings than in fuselage ( if the basic method of construction is similar) The effect of sheet panel shear strains is to cause some stringers to resist less or more axial load than those calculated by beam theory In general, the shear lag effect in skin-stringer box beam is not appreciable except for the following situations: Thin or soft (i. e. , aluminum) skin Cutouts which cause one or more stringers to be discontinued Large abrupt changes in external load applications Abrupt changes in stringer areas 3/2/2021 2

Shear Lag Example: Axial constraint stresses in a doubly symmetrical, single cell, six boom beam subject to shear. • The bending stress in box beams do not always conform very closely to the predictions of the simple beam bending theory. • The deviations from theory are caused primary by the shear deformations in the skin panels of the box that constitutes the flanges of the beam. • The problem of analyzing these deviations from the simple beam bending theory become known as the SHEAR LAG EFFECT 3/2/2021 3

Shear Lag Solution: Top cover of beam Loads on web and corner booms of beam Equilibrium of an edge boom element 3/2/2021 4

Shear Lag Similarly for an element of the central boom Now considering the overall equilibrium of a length z of the cover, we have 3/2/2021 5

Shear Lag We now consider the compatibility condition which exists in the displacement of elements of the boom and adjacent elements of the panel. or in which and are the normal strains in the elements of boom Now 3/2/2021 6

Shear Lag Choosing , say, the unknown to be determined initially. From these equations, we have Rearranging we obtain or Where is the shear lag constant The differential equation solution is The arbitrary constant C and D are determinate from the boundary conditions of the cover of the beam. 3/2/2021 7

Shear Lag when z = 0 ; when z = L From the first of these C = 0 and from the second Thus The normal stress distribution follows The distribution of load in the edge boom is whence 3/2/2021 8

Shear Lag The shear flow whence The shear stress Elementary theory gives 3/2/2021 9

Shear Lag Rectangular section beam supported at corner booms only The analysis is carried out in an identical manner to that in the previous case except that the boundary conditions for the central stringer are when z = 0 and z = L. Where 3/2/2021 is the shear lag constant 10

Shear Lag Beam subjected to combined bending and axial load 3/2/2021 11

Shear Lag is load in boom 1 is load in boom 2 Equilibrium of central stringer A Equilibrium of boom 2 Longitudinal equilibrium Moment equilibrium about boom 2 3/2/2021 12

Shear Lag The compatibility condition now includes the effect of bending in addition to extension as shown in figure below Where and z are function of z only Thus Similarly for an element of the lower panel Subtraction these equation or, as before 3/2/2021 13

Shear Lag Choose as the unknown, and using these equations we obtain or Where is the shear lag constant The differential equation solution is The arbitrary constant C and D are determinate from the boundary conditions of the cover of the beam. 3/2/2021 14

Shear Lag when z = 0 ; when z = L we have the distribution load in the central stringer or, rearranging The distribution of load in the edge booms 1 and 2 Finally the shear flow distribution are The shear flow and are self-equilibrating and are entirely produced by shear lag effect ( since no shear loads are applied). 3/2/2021 15