236 CHAPTER 6 Deflection of Beams

The method of superposition permits us to use the known displace-

ments and slopes for simple loads to obtain the deformations for more com-
plicated loadings. To use the method e¤ectively requires access to tables that
list the formulas for slopes and deflections for various loadings, such as
Tables 6.2 and 6.3. More extensive tables can be found in most engineering
handbooks.

Px 2 PL 3 PL 2
d¼ ð3L  xÞ dB ¼ yB ¼
6EI 3EI 2EI

8 2
>
> Px
>
< 6EI ð3a  xÞ 0 a x a a Pa 2 Pa 2
d¼ dB ¼ ð3L  aÞ yB ¼
>
> 2 6EI 2EI
: Pa ð3x  aÞ a a x a L
>
6EI

w0 x 2 w0 L 4 w0 L 3
d¼ ð6L 2  4Lx þ x 2 Þ dB ¼ yB ¼
24EI 8EI 6EI

8
>
> w0 x 2 2 2
>
< 24EI ð6a  4ax þ x Þ 0axaa
w0 a 3 w0 a 3
d¼ dB ¼ ð4L  aÞ yB ¼
>
> 3 24EI 6EI
: w0 a ð4x  aÞ
>
aaxaL
24EI

w0 x 2 w0 L 4 w0 L 3
d¼ ð10L 3  10L 2 x þ 5Lx 2  x 3 Þ dB ¼ yB ¼
120L EI 30EI 24EI

M0 x 2 M0 L 2 M0 L
d¼ dB ¼ yB ¼
2EI 2EI EI

TABLE 6.2 Deflection Formulas for Cantilever Beams

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Px L PL 3 PL 2
d¼ ð3L 2  4x 2 Þ 0 a x a d max ¼ yA ¼ yB ¼
48EI 2 48EI 16EI

8
> Pbx PbðL 2  b 2 Þ 3=2
>
> ðL 2  x 2  b 2 Þ 0axaa d max ¼ pffiffiffi
>
>
> 6L EI 9 3L EI Pab
< rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yA ¼ ð2L  aÞ
d¼ L2  b2 6L EI
Pb L
>
> ðx  aÞ 3 þ ðL 2  b 2 Þx  x 3 at x ¼ Pab
>
>
> 6L EI b 3 yB ¼ ð2L  bÞ
>
: 6L EI
aaxaL Pb
dcenter ¼ ð3L 2  4b 2 Þ if a > b
48EI

w0 x 5w0 L 4 w0 L 3
d¼ ðL 3  2Lx 2 þ x 3 Þ d max ¼ yA ¼ yB ¼
24EI 384EI 24EI

8
>
> w0 x w0
>
> ½a 2 ða  2LÞ 2 þ 2aða  2LÞx 2 þ Lx 3  dcenter ¼ ð5L 4  12L 2 b 2 þ 8b 4 Þ w0 a 2
>
> 384EI yA ¼ ða  2LÞ 2
< 24L EI 0axaa 24L EI
d¼ if a b b
>
> w0 a 2 w0 a 2
>
> ðL þ xÞða 2  4Lx þ 2x 2 Þ w0 a 2 yB ¼ ð2L 2  a 2 Þ
>
>
: 24L EI dcenter ¼ ð3L 2  2a 2 Þ if a a b 24L EI
aaxaL 96EI

w0 L 4
d max ¼ 0:006 522 7w0 L 3
EI yA ¼
w0 x 360EI
d¼ ð7L 4  10L 2 x 2 þ 3x 4 Þ at x ¼ 0:5193L
360L EI w0 L 3
w0 L 4 yB ¼
dcenter ¼ 0:006 510 45EI
EI
M0 L 2 L M0 L
d max ¼ pffiffiffi at x ¼ pffiffiffi yA ¼
M0 x 9 3EI 3 6EI
d¼ ðL 2  x 2 Þ
6L EI M0 L 2 M0 L
dcenter ¼ yB ¼
16EI 3EI

TABLE 6.3 Deflection Formulas for Simply Supported Beams

237

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.