# Shear and Moment Diagrams

Urban and Civil

Nov 15, 2013 (4 years and 6 months ago)

77 views

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 analysis

b) 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
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

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 clockwise

Moment:

+ imparts a U shape on section

Normal:

+ 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 = 0

M = 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

curve plus any concentrated forces

2) M(x) is the area under V(x)

This relationship is proven in your text

gets you the diagrams quicker

derivation assumes

Examine a diff beam

section

Example3

Reactions at B