SHEAR LAG

1. SHEAR LAG Prof. Carlos Montestruque

2.  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 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)  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. 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 the 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

4. Solution: Top cover of beam Equilibrium of an edge boom element Loads on web and corner booms of beam

5. Similarly for an element of the central boom Now considering the overall equilibrium of a length z of the cover, we have

6. 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

7. 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.

8. 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

9. The shear flow whence The shear stress Elementary theory gives

10. 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 is the shear lag constant

11. Beam subjected to combined bending and axial load

12. is load in boom 1 Equilibrium of boom 2 is load in boom 2 Equilibrium of central stringer A Equilibrium of boom 2 Longitudinal equilibrium Moment equilibrium about boom 2

13. 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

14. 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.

15. 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).