Shear and Moment Diagrams

reelingripebeltΠολεοδομικά Έργα

15 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

71 εμφανίσεις

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
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 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


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