Shear and Moment Diagrams

Today’s Objective:Students will be able to:1. Derive shear and bending moment

diagrams for a loaded beam using a) piecewise analysisb) differential/integral relations

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

General Technique• Because the shear

and bending moment are discontinuous near a concentrated load, they need to be analyzed in segments between discontinuities

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

SIGN CONVENTION FOR SHEAR, BENDING MOMENT

Sign convention for:

Shear: + rotates section clockwiseMoment: + imparts a U shape on sectionNormal: + creates tension on section(we won't be diagraming nrmal)

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

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 = 0M = 10 - 2.5x

Example, continued

• Now, plot the curves in their valid regions:

• Note disconinuities due

to mathematical ideals

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

Example2, (cont)• 1) Reactions on beam• 2) FBD of left section

– note sign convention

• 3) Solve:

Example 2, continued

• Plot the curves:

• Notice Max M occurs• when V = 0?

• could V be the slope of M?

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

derivation assumes positive distrib load

Examine a diff beam section

Example3

• Reactions at B