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B

en

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ng

III

:

Deflection of Beams

MAE 314 –Solid Mechanics

Y. Zhu

Introduction

•Up to now, we have been primarily calculating normal and

shear stresses.

•In this lecture, we will learn how to formulate the deflection

curve(also known as the elastic curve) of a beam to due

transverseloading.

transverse

loading.

Slide 2

Differential Equation of Deflection

ds

d

=

θ

ρ

d

θ

tan

d

y

ds

d

=

θ

ρ

θ

1

=

d

d

dy

d

s

θ

θ

tan

=

dx

y

dx

ρ

d

s

Recall from Ch. 4 that

1/ρis the curvature of

y

dx

θ

θ

cos

=

ds

dx

θ

sin

=

摹

the beam.

θ

sin

=

ds

Slope of the deflection curve

Slide 3

Assumptions

•Assumption 1: θis small.

–

1.

⇒

≅

d

d

dd

θ

θ

≅

1

–

2

⇒

≅

d

x

d

s

dxds

ρ

≅

=

⇒

≅

=

θ

θ

tan

dy

⇒

=

2

ydd

θ

2

1yd

=

–

2

.

•

A

ssum

p

tion 2: Beam is linearl

y

elastic.

⇒

≅

θ

θ

tan

dx

⇒

=

2

dxdx

2

dx

=

ρ

M

1

py

•Thus, the differential equation for the deflection curve is:

EI

=

ρ

EI

M

dx

yd

=

2

2

Slide 4

Diff. Equations for M, V, and w

•Recall :

w

dx

dV

−=

V

dx

dM

=

•So we can write:

w

dx

yd

EI−=

4

4

V

dx

yd

EI=

3

3

M

dx

yd

EI=

2

2

•Deflection curve can be found by integrating

dx

dx

dx

–Bending moment equation (2constants of integration)

–Shear-force equation (3constants of integration)

–Load equation (4constants of integration)

•Chosen method depends on which is more convenient.

Slide 5

Boundary Conditions

•Sometimes a single equation is sufficient for the entire length of the

beam, sometimes it must be divided into sections.

Siitttithillb

t

ttfittif

•

Si

nce we

i

n

t

egra

t

e

t

w

i

ce

th

ere w

ill

b

e

t

wocons

t

an

t

s o

f

i

n

t

egra

ti

on

f

or

each section.

•These can be solved using boundary conditions.

Dflidl

–

D

e

fl

ect

i

ons an

d

s

l

opes at supports

–Known moment and shear conditions

Slide 6

Boundary Conditions cont’d

•Continuity conditions:

–Displacement continuity

Section AC: y

AC(x)Section CB: y

CB(x)

–Slope continuity

)()(C

y

C

y

CBAC

=

)

(

)

(

)

(

)

(

C

C

dy

C

C

dy

CBAC

θ

θ

=

=

=

摹

•Symmetry conditions:

)

(

)

(

)

(

)

(

C

C

dx

C

C

dx

CB

AC

θ

θ

=

=

=

0

=

dx

dy

Slide 7

Example Problem 1

The cantilever beam AB is of uniform cross section and carries a load

P at its free end A. determine the equation of the elastic curve and the

deflection and slop at A (Example 901 in Beer

’

s book (P535))

deflection and slop at A (Example 9

.

01 in Beers book (P535))

.

Slide 8

Slide 9

Example Problem 2

For the beam and loading shown, (a) express the magnitude and

location of the maximum deflection in terms of w

0, L, E, and I.

(b)Calclatethealeofthemaimmdeflectionassmingthatbeam

(b)

Calc

u

late

the

v

al

u

e

of

the

ma

x

im

u

m

deflection

,

ass

u

ming

that

beam

AB is a W18 x 50 rolled shape and that w

0

= 4.5 kips/ft, L = 18 ft, and

E = 29 x 10

6

psi.

Slide 10

Slide 11

Statically Indeterminate Beams

•When there are more reactions than can be solved using

statics, the beam is indeterminate.

Tkdtfbddititlidtit

•

T

a

k

e a

d

van

t

age o

f

b

oun

d

ary con

diti

ons

t

o so

l

ve

i

n

d

e

t

erm

i

na

t

e

problems.

Pr

ob

l

e

m:

obe

Number of reactions: 3 (M

A, Ay

, By)

Number of equations: 2 (ΣM = 0, ΣFy = 0)

x=0, y=0

x=0, θ=0

x=L, y=0

One too many reactions!

Additionally, if we solve for the deflection curve,

we will have two constants of inte

g

ration, which

g

adds two more unknowns!

Solution:Boundary conditions

Slide 12

Statically Indeterminate Beams

Problem:

Number of reactions: 4 (M

A, Ay

, MB, By)

Number of equations: 2 (ΣM = 0, ΣFy = 0)

+2constantsofintegration

x=0, y=0

x=0, θ=0

x=L, y=0

x=0, θ=0

+

2

constants

of

integration

Solution:Boundary conditions

Slide 13

Example Problem 3

•Determine the reactions at the supports for the prismatic

beam shown below

(

Exam

p

le 9.05 in Beer’s book

)

.

(p)

Slide 14

Slide 15

Example Problem 4

For the beam shown determine the reaction at the roller support when

w

0

= 65 kN/m.

0

Slide 16

Slide 17

Method of Superposition

•Deflection and slo

p

e of a beam

p

roduced b

y

multi

p

le loads

ppyp

acting simultaneously can be found by superposingthe

deflections produced by the same loads acting separately.

•Reference Appendix E in Craig’s book (Beam Deflections and

Slopes)

•Method of superposition can be applied to statically

determinate and statically indeterminate beams.

Slide 18

Superposition cont’d

•Consider the

p

roblem on the ri

g

ht.

pg

•Find reactions at A and C.

•Method 1: Choose M

C

and RC

as

redundant.

•

Method2:ChooseM

andM

asredundant

Slide 19

•

Method

2:

Choose

M

C

and

M

A

as

redundant

.

Example Problem 5

For the beam and loading shown, determine (a) the deflection at C, and

(

b

)

the slo

p

e at end A.

()p

Slide 20

Slide 21

Example Problem 6

For the beam shown, determine the reaction at B.

Slide 22

Slide 23

Example Problem 4

The overhanging steel beam ABC carries a concentrated load P at end C.

For portion AB of the beam, (a) derive the equation of the elastic curve,

(b) determine the maximum deflection (c) evaluate y

for the following

(b) determine the maximum deflection

,

(c) evaluate y

max

for the following

data:

W 14x68I = 723 in

4

E = 29x106

psi

P = 50 kipsL = 15 ft = 180 in.a = 4 ft = 48 in.

(sample problem 9.1 in Beer’s book (p. 542))

Slide 24

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