Dr. Ing. Zdenka Sant Mechanics of Material I 7

Bending stress and deflection

1. Derive the formula for evaluation of the maximum bending moment Mmax that can be carried by each of the cross sections shown in the figure, if the allowable stress (tension or compression) is w?

b b b d h

b 2b 2d

Ans.: 32

.

3

max

bM w= ; 6013

.

2

max

bhM w= ; 6415

.

3

max

dM wpi

=

2. Determine the maximum normal stress due to bending for the simple beam shown in the figure, if F=5kN and the cross section has the dimensions given in the sketch. Find the deflection at the mid-span. Ans.: =121MPa

1200mm 75 F A B 75 25

3000mm 25

3. A simply supported beam with a rectangular cross section supports a uniformly distributed load that deflects the beam 2mm at the center. This beam is to be replaced by another beam of the same material and also of rectangular cross section, but with only half width of the original beam. What must be the height of the new beam compared to the height of the original beam if the new beam is to deflect only 0.5mm under the same load? Ans.: hn=2h

4. Derive the equation of the deflection curve for a simple beam loaded by a couple Mo acting at the distance a from the left end.

Mo A B a L

Dr. Ing. Zdenka Sant Mechanics of Material I 8

5. The beam shown in the figure supports two concentrated loads. Find the angle of rotation 1, the deflection 1 under the load F1 and the deflection 2 at the mid-span.

a F1 F2 a

A B

L

6. Derive an expression for the max. deflection of a simply supported beam with negligible weight carrying concentrated load at its mid-span position. The distance between supports is L, the second moment of area of the cross section is I and the modulus of elasticity of the beam material is E.

7. The maximum deflection of such a simply supported beam of length 3m is 4.3mm when carrying a load of 200kN at its mid-span position.

What would be the deflection at the free end of a cantilever of the same material, length and cross section area if it carries a load of 100kN at a point 1.3m from the free end? [=0,0134m]

8. A beam 7m long is simply supported at its ends and carries a load as follows:120kN at 1m from end A, 20kN at 4m from A and 60kN at 5m from A.

Calculate the position and magnitude of the maximum deflection. The second moment of area of the beam section is 400.10-6m4 and E=2,1.105MPa. 1m 20kN

120kN 60kN A B 4m 1m L

9. A beam with length L, is simply supported at its ends and carries a uniformly distributed load of q per unit length between mid-span and a point at the distance L/4 from the mid-span on the right-hand end.

10. A horizontal beam freely supported at its ends on two similar cantilevers is loaded by two concentrated forces acting at the L/3 distance from supports. If I, the second moment of area of the cantilever section, is twice that for the beam, derive an expression, in terms of F, L,E and I, for the deflection at the centre of the beam.

11. Evaluate the deflection at the mid-span for beams from tutorial sheet Stress resultants and Moments of inertia assuming the beam cross-section shape:

a) rectangular cross-section b) T-shape cross-section c) Right angle triangle