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Shear and Moment Diagrams

Shear and Moment Diagrams. Today’s Objective : Students will be able to: Derive shear and bending moment diagrams for a loaded beam using a) piecewise analysis b) differential/integral relations. These diagrams plot the internal forces with respect to x along the beam. APPLICATIONS.

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Shear and Moment Diagrams

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  1. Shear and Moment Diagrams • Today’s Objective: • Students will be able to: • Derive shear and bending moment diagrams for a loaded beam using a) piecewise analysis • b) differential/integral relations

  2. These diagrams plot the internal forces with respect to x along the beam. APPLICATIONS They help engineers analyze where the weak points will be in a member

  3. General Technique • Because the shear and bending moment are discontinuous near a concentrated load, they need to be analyzed in segments between discontinuities

  4. Detailed Technique • 1) Determine all reaction forces • 2) Label x starting at left edge • 3) Section the beam at points of discontinuity of load • 4) FBD each section showing V and M in their positive sense • 5) Find V(x), M(x) • 6) Plot the two curves

  5. SIGN CONVENTION FOR SHEAR, BENDING MOMENT Sign convention for: Shear: + rotates section clockwise Moment: + imparts a U shape on section Normal: + creates tension on section (we won't be diagraming nrmal)

  6. Example • Find Shear and Bending • Moment diagram for the beam • Support A is thrust bearing (Ax, Ay) • Support C is journal bearing (Cy) • PLAN • 1) Find reactions at A and C • 2) FBD a left section ending at x where (0<x<2) • 3) Derive V(x), M(x) • 4) FBD a left section ending at x where (2<x<4) • 5) Derive V(x), M(x) in this region • 6) Plot

  7. Example, (cont) • 1) Reactions on beam • 2) FBD of left section in AB  • note sign convention • 3) Solve: V = 2.5 kN M = 2.5x kN-m • 4) FBD of left section ending in BC: • 5) Solve: V = -2.5 kN -2.5x+5(x-2)+M = 0 M = 10 - 2.5x

  8. Example, continued • Now, plot the curves in their valid regions: • Note disconinuities due to mathematical ideals

  9. Example2 • Find Shear and Bending • Moment diagram for the beam • PLAN • 1) Find reactions • 2) FBD a left section ending at x, where (0<x<9) • 3) Derive V(x), M(x) • 4) FBD a left section ending somewhere in BC (2<x<4) • 5) Derive V(x), M(x) • 6) Plot

  10. Example2, (cont) • 1) Reactions on beam • 2) FBD of left section  • note sign convention • 3) Solve:

  11. Example 2, continued • Plot the curves: • Notice Max M occurs • when V = 0? • could V be the slope of M?

  12. A calculus based approach • Study the curves in the previous slide • Note that • 1) V(x) is the area under the loading curve plus any concentrated forces • 2) M(x) is the area under V(x) • This relationship is proven in your text • when loads get complicated, calculus gets you the diagrams quicker

  13. derivation assumes positive distrib load Examine a diff beam section

  14. Example3 • Reactions at B

  15. End of the Lecture Let Learning Continue

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