Bending 6.3-6.4

1. Bending 6.3-6.4 Longitudinal Strain Flexure Formula

2. Bending Theory There are a number of assumptions made in order to develop the Elastic Theory of Bending: • The beam has a constant cross-section • Is made of a flexible, homogenous material with the same E in tension and compression. • The material is linearly elastic. Stress and Strain are proportional (material obeys Hooke’s Law). • The beam material is not stressed past its proportional limit.

3. Longitudinal Strains in Beams • Approach: • The Longitudinal Strains in beam can be determined by analyzing the curvature of the beam and its deformations. • Consider a portion of a beam subjected only to positive bending. • Assume that the beam is initially straight and is symmetrical.

4. Longitudinal Strains in Beams • Under the action of the bending moments, the beam deflects in the xy plane and its longitudinal axis is bent into a curve…positive. • mn and pq remain plane and normal to the longitudinal axis. • This fact is fundamental to beam theory. • Even though a plane cross section in pure bending remains plane, there still may be deformations in the plane itself, but they are small and may be neglected.

5. Longitudinal Strains in Beams • With the beam in positive bending, we know that the lower part of the beam in is Tension and the upper part is in compression. • Somewhere between the top and bottom of the beam is a surface that does not change in length. This is called the neutral surface of the beam.

6. Longitudinal Strains in Beams • The section mnpq intersect in a line through the center of curvature O, through an angle d, at a radius of . Figure 6.23 • At the neutral surface, dx is unchanged, therefore d=dx. • All other longitudinal lines between the two planes either lengthen or shorten, creating normal strains x.

8. Longitudinal Strains in Beams • To evaluate these normal strains, consider line ∆s, a distance y from the neutral surface. • (we are assuming that the x axis lies along the neutral surface of the undeformed beam. When the beam deflects, the neutral surface moves with the beam, but the x axis remains in a fixed position. Yet Δs in the deflected beam is still located at the same distance y from the neutral surface). • Therefore the length of ef after bending is:

9. Longitudinal Strains in Beams • Since the original length Δs is dx, it follows that its elongation is Δs-dx or –ydx/ . • The longitudinal strain is the elongation divided by the initial length dx. • This is called the Strain-Curvature Relation.

11. Longitudinal Strains in Beams • The equation was derived solely from the geometry (without concern for material). • Strains in a beam in pure bending vary linearly with distance from the neutral surface regardless of the shape of the stress-strain curve of the material. • Longitudinal strains in a beam are accompanied by transverse strains because of Poisson’s ratio.

12. Longitudinal Stresses in Beams • Now we need to find the stresses from the strains, using a stress-strain curve. • Note there are no accompanying transverse stresses because beams are free to deform laterally.

13. Normal Stresses in Beams • Stresses act over the entire area cross section of the beam and vary in intensity depending upon the shape of the stress-strain diagram and the dimensions. Fig 6.26 • For a linearly elastic material, we can use Hooke’s law

15. Normal Stresses in Beams • By substitution we get: • This equation shows that the normal stresses acting on the cross section vary linearly with the distance from y from the neutral surface. Fig 6.26b.

16. Normal Stresses in Beams • In general, the resultant of the normal stresses consists of two stress resultants: • A force acting in the x-direction • A bending couple acting about the z axis. • Since the axial force is zero for a beam in pure bending: • The resultant force in the x direction is zero (this gives us the location of the neutral axis) • The resultant moment is the bending moment.

17. Moment-Curvature Relationship • Therefore the neutral axis passes through the centroid of the cross-sectional area for our given constraints. • And the moment resulting from the normal stresses acting over the cross section is equal to the bending moment.

18. Moment-Curvature Relationship • The integral of all such elemental moments over the entire cross-sectional area is then equal to the bending moment. Equation 6-11 • Recall that is the moment of inertia. • Then we can express curvature in terms of the bending moment called the Moment Curvature Equation.

19. Moment-Curvature Relationship • This shows that the curvature is directly proportional to the bending moment applied and inversely proportional to EI – called the Flexural Rigidity of the beam. • Flexural Rigidity is a measure of the resistance of a beam to bending.

20. Flexure Formula • Now that we have located the neutral axis and derived the moment curvature relationship we can find the stresses in terms of bending moment. • If we substitute the equation for curvature into our equation for normal stress we get • Called the Flexure Formula.

21. Flexure Formula • Stresses calculated with the flexure formula are called Bending Stresses or Flexural Stresses. • The maximum tensile and compressive bending stresses acting at any given cross section occur at points farthest from the neutral axis.

22. Limitation Reminders • The beam is subjected to Pure Bending • The beam has a constant cross-section • Is made of a flexible, homogenous material with the same E in tension and compression. • The material is linearly elastic. Stress and Strain are proportional. • The beam material is not stressed past its proportional limit.