Beams Shear & Moment Diagrams

1. BeamsShear & Moment Diagrams E. Evans 2/9/06

2. Beams • Members that are slender and support loads applied perpendicular to their longitudinal axis. Concentrated Load, P Distributed Load, w(x) Longitudinal Axis Span, L

3. FH Pin FV FV Fixed FH Roller M Fv Roller Pin FH FV FV Types of Beams • Depends on the support configuration

4. Statically Indeterminate Beams • Can you guess how we find the “extra” reactions? Continuous Beam Propped Cantilever Beam

5. P a b L Internal Reactions in Beams • At any cut in a beam, there are 3 possible internal reactions required for equilibrium: • normal force, • shear force, • bending moment.

6. M N V Internal Reactions in Beams • At any cut in a beam, there are 3 possible internal reactions required for equilibrium: • normal force, • shear force, • bending moment. Positive Directions Shown!!! Left Side of Cut Pb/L x

7. V M N Internal Reactions in Beams • At any cut in a beam, there are 3 possible internal reactions required for equilibrium: • normal force, • shear force, • bending moment. Positive Directions Shown!!! Right Side of Cut Pa/L L - x

8. Finding Internal Reactions • Pick left side of the cut: • Find the sum of all the vertical forces to the left of the cut, including V. Solve for shear, V. • Find the sum of all the horizontal forces to the left of the cut, including N. Solve for axial force, N. It’s usually, but not always, 0. • Sum the moments of all the forces to the left of the cut about the point of the cut. Include M. Solve for bending moment, M • Pick the right side of the cut: • Same as above, except to the right of the cut.

9. Example: Find the internal reactions at points indicated. All axial force reactions are zero. Points are 2-ft apart. P = 20 kips 1 2 3 4 5 8 9 10 6 7 8 kips 12 kips 12 ft 20 ft Point 6 is just left of P and Point 7 is just right of P.

10. P = 20 kips 1 2 3 4 5 8 9 10 6 7 8 kips 12 kips 12 ft 20 ft 8 kips V (kips) x -12 kips 96 80 72 64 48 48 32 24 16 M (ft-kips) x

11. What is the slope of this line? What is the slope of this line? V & M Diagrams P = 20 kips 8 kips 12 kips 12 ft 20 ft 8 kips V (kips) x -12 kips 96 ft-kips 96 ft-kips/12’ = 8 kips b -12 kips M (ft-kips) a c x

12. V & M Diagrams P = 20 kips 8 kips 12 kips 12 ft 20 ft 8 kips V (kips) x What is the area of the blue rectangle? -12 kips 96 ft-kips What is the area of the green rectangle? 96 ft-kips b -96 ft-kips M (ft-kips) a c x

13. Draw Some Conclusions • The magnitude of the shear at a point equals the slope of the moment diagram at that point. • The area under the shear diagram between two points equals the change in moments between those two points. • At points where the shear is zero, the moment is a local maximum or minimum.

14. The Relationship Between Load, Shear and Bending Moment

15. Common Relationships

16. Common Relationships M

17. Example: Draw Shear & Moment diagrams for the following beam 12 kN 8 kN A C D B 1 m 3 m 1 m RA = 7 kN  RC = 13 kN 

18. 12 kN 8 kN A C D B 1 m 3 m 1 m 8 7 8 7 V (kN) -15 -5 7 M (kN-m) 2.4 m -8