Transverse Shear

1. Transverse Shear Shear in Straight Members Shear Formula Shear Stresses in Beams

2. Shear Stresses in Beams of Rectangular Cross Section • In the previous chapter we examined the case of a beam subjected to pure bending i.e. a constant moment along axis . • When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections. • Most beams are subjected to loads that produce both bending moments and shear forces (non-uniform bending)

3. Shear Stresses in Beams of Rectangular Cross Section • In these cases, both normal and shear forces are developed in the beam. • Normal stresses are calculated with the Flexure Formula. • We will now look at the Shear Stresses

4. Vertical & Horizontal Shear Stresses • Consider a beam of rectangular cross section subjected to a positive shear force.

5. Vertical & Horizontal Shear Stresses • We assume • That the shear stresses are parallel to the shear force • That the shear stresses are uniformly distributed across the width of the beam. • Using these assumptions we can determine the intensity of the shear stresses at any point on a cross section.

6. For analysis we isolate a small element We determined back in section 1.5 that there are horizontal shear stresses acting between horizontal layers of the beam as well as vertical shear stresses acting on the cross section. At any point in the beam, these complementary shear stresses have the same magnitude. Vertical & Horizontal Shear Stresses

7. Vertical & Horizontal Shear Stresses • Consider a cantilever beam made up of planks that are not bonded and subjected to transverse loading. • Each individual plank will be subjected to slip at the interfaces. • Plane sections before deformation do not remain plane after deformation.

8. If the planks are bonded together to form a single beam, longitudinal shear stresses must develop to prevent the relative sliding between layers. There is maximum shear distortion at the middle plank, while distortion is zero at the top and bottom of the beam Vertical & Horizontal Shear Stresses

9. Vertical & Horizontal Shear Stresses • Because of shear deformation, plane sections do not remain plane. • This is in contradiction with theory of pure bending in which plane sections were assumed to remain plane after deformations. • So, we will assume that the shear deformations have little effect on the distribution of flexural stresses as long as the beam is slender, say length at least 10 times the depth.

10. Vertical & Horizontal Shear Stresses Also the following assumptions will be retained, i.e. ·       The transverse shear strains are not simple to determine and will be ignored for a long slender beam. ·       The shear stresses will only be determined from equilibrium rather than from the strains as before.

11. Derivation of Shear Formula • Now we can derive a formula for the shear stresses in a rectangular beam. • Instead of evaluating the vertical shear stresses acting on a cross section, it is easier to evaluate the horizontal shear stresses acting between layers of the beam. • Remember, the vertical shear stresses have the same magnitude as the horizontal shear stresses.

12. Derivation of Shear Formula • Consider a beam with non-uniform bending. • Take two adjacent cross sections a distance dx apart. • The bending moment and shear force acting on the left side is M and V. • Since the moment and shear force may change along the axis of the beam, the right side is noted as M+dM and V +dV.

13. Derivation of Shear Formula • The normal stresses on the section are found with the Flexure Formula. • Next if isolate a sub-element by passing a horizontal plane through the element.

14. Because the bending moments vary along the x- axis (non-uniform bending), we can determine the shear force acting on the bottom surface of the sub-element by equilibrium. The force acting on this element is dA  is the normal stress obtained from the flexure formula On the left side then: The total force over the are of the face of the sub-element is: Derivation of Shear Formula

15. Derivation of Shear Formula • The Force acting on the right side would be found similarly. • Knowing F1 and F2, we can then find the force on the bottom of the sub-element, by equilibrium (see equation 7-1) • The integral in the final equation is evaluated over the shaded part of the cross section. • This is called the first moment of area.

16. Shear Formula • Solving for  yields the Shear Formula • The shear formula can be used to determine the shear stress at any point in the cross section of a rectangular beam. • Note • for a specific cross section, the shear force V, moment of inertia I, and width t are all constants. • Q – the first moment of area varies with the distance from the neutral axis.

17. Calculation of Q • The first moment of area can be calculated for the cross-section above the neutral axis +Q or below the neutral axis -Q. • The first moment of the entire cross section with respect to the neutral axis is zero. • However, we usually don’t bother with sign conventions for V and Q. • We treat all terms in the shear formula as positive and determine the direction of shear stresses by inspection.

18. Distribution of Shear Stresses in a Rectangular Beam • We can now determine the distribution of the shear stresses in a beam that has a rectangular cross section. • Substituting the expression for Q into the shear formula yields

19. Distribution of Shear Stresses in a Rectangular Beam • The shear stresses in a rectangular beam vary quadratically with the distance from the neutral axis. • Note that the shear stress is zero when y1=± h/2 • When plotted along the height of the beam, they would look like:

20. Distribution of Shear Stresses in a Rectangular Beam • The maximum value of shear stress occurs at the neutral axis (y1=0) the first moment Q has a maximum value. • Substituting y1=0 into the previous equation we get:

21. Distribution of Shear Stresses in a Rectangular Beam • Limitations • Same restrictions as the flexure formula. • Valid only for beams of linearly elastic materials with small deflections. • In rectangular beams, the accuracy depends on the width to height ratio of the cross section. • Exact for very narrow beams (h much larger than b) • When b=h, the true maximum shear stress is about 13% larger than the value given by the formula.

22. Distribution of Shear Stresses in a Rectangular Beam • A common error is to apply the shear formula to cross sectional shapes for which it is not applicable. • It is not applicable to triangular or semicircular shapes. • Applies only to prismatic beams (constant cross section) • Edges of the cross section must be parallel to the y axis • Shear stresses must be uniform across the width of the cross section.

23. Stresses in thin-walled sections • Thin-walled section beams are beams with sections for which the wall thickness is significantly smaller than the overall dimensions of the cross section of a tubular beam, or the depth of a wide-flange beam. • When dealing with thin walled beams, we can consider the following: • Shear stress distribution is uniform across the thickness of the wall. • Shear stresses can be approximated as being parallel to the wall axis.

24. Shear stress in web-flange beams • Sections (web and flanges) are much more efficient than compact sections such as rectangular sections in their resistance to bending (for the same area) • Basically efficiency refers to strength to weight. • We will now examine the shear stress distribution in the section, i.e. in the flanges and the web by making appropriate cuts and exposing the internal forces.

25. Shear stress in web-flange beams • As shown in the FBD, a shear force dF acting on the cutting plane is required to balance the flexural stresses acting on each side of the section of elemental length dx. • The shear flow is defined as and is dictated by the direction of dF in the longitudinal plane of cut.

26. Shear stress in web-flange beams • Thus: • It is noted that only shear flow parallel to the wall is considered, and the one across the width t is zero. • The term Q refers to the first moment area of shaded area about neutral axis.

27. Shear stress in web-flange beams • The variation of shear flow across the section depends only on the variation of the first moment • For a box beam, q grows smoothly from zero at A to a maximum at C and C’ and then decreases back to zero at E.

28. Shear stress in web-flange beams • For a wide-flange beam, the shear flow increases symmetrically from zero at A and A', reaches a maximum at C and the decreases to zero at E and E‘ • See Figure 7-7c