Appendix Tables of Beam Deflections978-94-011-6535-8/1.pdf · TABLES OF DEFLECTIONS OF STATICALLY...
15
Appendix Tables of Beam Deflections Statically Determinate Beams 404-409 Statically Indeterminate Beams 410-412
Transcript of Appendix Tables of Beam Deflections978-94-011-6535-8/1.pdf · TABLES OF DEFLECTIONS OF STATICALLY...
Appendix
Tables of Beam Deflections
Statically Determinate Beams 404-409
Statically Indeterminate Beams 410-412
TAB
LES
OF
DEF
LEC
TIO
NS
OF
STA
TIC
ALL
Y D
ETER
MIN
ATE
BEA
MS
g A
PP
EN
DIX
A
Bea
m, l
oadi
ng, a
nd
diag
ram
s of
mom
ents
A
ngle
s of
Su
ppor
t an
d sh
ear
forc
es
Elas
tic c
urve
ro
tatio
n B
endi
n!!
mom
ents
Sh
ear
forc
es
reac
tions
W=
PI3 [~-.
!£+
ptZ
1[
Sim
:~Y Su
ppor
ted 1s
N=
-P [
t-lI
fl]
Rl
=-~
I Concentra~loadin
w'=
16E
/ 1
-4[2
+
M=
-PI
21-
the
mid
-ClO
ss-s
ectio
n 16
E/
I 3
13
2
+II,!
[~-.!.r
] +
IIf
8 [f-trJ
-lif
[f-t]
J
Rt~~~1:
"23
I 2
PI3
Plz
M ...
. =-
PI
Rz=-
~ ~
Wm
ox=
48E
I w
o= 1
6EI
4 2
'-'i:N
fo
n=
.!.
for
x=O
fo
r x=
.!.
L~ _
_ .
J
2 2
Plz
w,=
-16
EI
for
x=1
Concentra~ lo
ad in
W
=:
[e;·
f [1
<:
-w
'eP
I' [~[l-~-
M=-Pll
-~:-IL
[x7]J
N=
-P [
e; -
ILl]
e'
2
R1
=-P
T
the
mid
-cro
ss-s
ectio
n 6E
J 1
I'
-;:]
+11
. [x
;er]
-3~:
]+Ib
[7]']
Rtl;;g;jR~
pc2
c'2
, Pe
e' [
e']
Pee
' R
z=-P
!:..
Y
N
1 W
,=
3E1l
wO
=6E
1 1
+/
Mm
ox=
--I-
~
I --
----
-+- L_
.J
for
x=e
for
x=O
fo
r x=
e P
e'lz
,
Pee
' [
e]
wm
ox=O
.064
1 E
/x
W,=
6EI
1
+/
e"2
"2
[ r
x 1
-[2
fo
r x=
1
for
x=-V
e-,
/"2
,
if e>
e'
. _
--
_._
--------------_
.
-.. _
-
In a
ll th
e ta
bles
, th
e te
rms
afte
r th
e si
gn II,
shou
ld b
e co
nsid
ered
onl
y fo
r cr
oss-
sect
ions
x >
c.
Con
l'd
Bea
m. l
oadi
ng. a
nd
diag
ram
s of
mom
ents
Ang
les
of
Supp
ort
and
shea
r fo
rces
El
astic
cur
ve
rota
tion
Bend
inR
mom
ents
Shea
r fo
rces
re
actio
ns
3 Tw
o sy
mm
etric
for
ces
w=~ [~
[3~--~~+
w'=
PIZ
l_e
e' _
xZ
+
{x
II x-c
N
=-P
(I -
1J.
1-I
I.,I
) M
=-P
I --
---
Rt=
Rz=
-P
in th
e sp
an
6IiJ
/
/'
/'
2EJ
12
12
I •
I
+IL[
x~er
+ +I
L[x~
cr+
_II ,x
-c']
•
I
R,t
t ~c
~R~
+IL[x~
eT]
+IL[x~
eT]
1L-c
....!-
---c
'---
---f
-
JY-
"'N
M
._ . .
L.._
. __
__
w
= Pic
' [3'£
'-E.]
Pce
' M
....
=-P
c
L _
_
'6E
1
I I
WO
=-W
I=W
w f:
Xp~;c
[~_~
] fo
r c:
!>x
:!>c'
-6E
I 4
I'
forx
=i 2
4 C
once
nuat
ed m
orne
nt a
t a
W=
-:;
[f-
;:-J
w
'=-
HI
[1-3
X2-J
M=
H!.
N
=!!
. R
I =
+!!
. su
ppor
t cro
ss-s
ecuo
n m~.
6E/
12
I I
I
HI'
H
I R
2=
-!!.
.. ~
w_
=-0
_077
8"E
J w
o=
-6E
J M
mu
=H
I
for x
= 0_
5771
H
I fo
r x=
1 W
,= 3
El
5 Tw
o co
ncen
uate
d m
omen
ts w
=-:~
f [l-f]
x w'=-~ [~
[f-
M=
M' [
I-f)
+H
"f
11'-
11"
H
'-M
" N
=----
R1
=---I-
at s
uppo
rt cr
oss-
sect
iorts
I
x [M'[
2-f
]+
x X
2]
-2
-+
-+
ftL:\~
1 12
+H
"[I+
f]]
+H
" [+-
;: ]]
M=
M' f
orx=
O
M'-
H"
R2=
--I-
, ,M
,
2M
'+M
" 1
6i=_~
wo
=--w
-M
=M
"fo
rx=
1
i ,
JI'+
2H"
W
I=--w
-I
§
6 7 8
Bea
m, l
oadi
ng,
and
diag
ram
s of
mom
ents
an
d sh
ear f
orce
s
Con
cent
rate
d m
omen
t in
the
spa
n
R·t;
Qc'=
4~
C t
R.
V~
Uni
form
ly d
islri
bute
d lo
ad o
ver
entir
e sp
an
R~Q=qt R
. q
•
t Y
M
~~
~--=---~
Uni
form
ly d
islri
bute
d lo
ad
over
a p
onio
n of
a s
pan
R.
Ela
stic
cur
ve
w~MI
' [~[1-3~-
6E/
I I'
-1.+1 1.3
[x; r]
we=
~~~'
[c/]
QI'
[x
x' X4
] w
= 2
4EI
1-2
,,+
7
5 Q
I' W
"",.=
384
m
1 fo
r x=
"2
QI'c'
[x
[ c
w~
24E
l 7
1+27
-
_£.-
2~]+
1I (X
-C)4]
12
12
c
Pc"
Ang
les
of
rota
tion
W'=
III
[1
-3~-
6£1
I'
-31,-+
11.6[7
]] ,
III
[ c,,]
W
.= 6
E/
1-3
/2
, II
I [
c']
w,=
6E
l 1
-3
"
, II
I [cc
' ]
we=
3E
l 3
/2-1
,-~ 1
-6,+
,[
x'
w-
24E
l I
x']
+4
- I'
, 'Q
12
wo=
-w,=
24E
l
w'=
QIc
' [1
+2~-
24E
l I
c2 .x
2 +
-.-6
I'
I II (X
-C)']
+
,4
Ie
"
, Q
lc'
x W
o= 2
4EI
[ c_
d x
1+21
I'
J
,
, Q
lc' rl
+~1
w,=
-24
1':1
l I
J
Ben
dine
mom
ents
M=
-M[f
-lL1 ]
M=
-Q
I [~_
.£.]
2 1
I'
QI
M ..
.. =
-g=
-O.l
25
QI
for
x= i 2
M=
-Q
I 2
M""
,.=
-Q
c'
[ c]'
8 1
+- 1
Shea
r fo
rces
N=-!!
. 1
N=-%
[1-~
]
N=-Q
[£-1
1 ~]
21
'
, c
Con
t'd
Sup
pon
reac
tions
Rl=
-!:!
. 1
R2=
!:!.
1
Rl=
R2
=-Q
2
R1=
-Q
c'
21
R2=-Q
[1-ft
]
Con
t'd
Bea
m,
load
ing,
and
di
agra
ms
of m
omen
ts
Ang
les
of
Supp
ort
and
shea
r fo
rces
E
last
ic c
urve
ro
tatio
n B
endi
ng m
omen
ts
Shea
r fo
rces
re
actio
ns
9 L
inea
rly
dist
ribu
ted
QI'
[X
x'
w'=~x
M=
-~I
[f-[fn
N=
-Q [
1-3
£]
Rl=
-Q
load
ove
r en
tire
span
w
=
180E
I 7
/-1
01
'+
180E
I 3
12
3
x5
] [
x2 x']
+
3-
x 7
-30
-+1
5-
'.~~
15
12
I'
QI3
_
' 7
QI2
M
"". =
-0.1
283Q
I w
mu
=0.
0130
4 E
I-w
o=T
sO E
I=
_ 5
QI3
_
QI2
fo
r x=
0.57
71
2 -
384 E
I -0
.038
9E
I R
2=-3
"Q
-....
<.,NJ
fo
r x
=0.
5191
'
2 Q
12
_
w,=
-45
EI-
_ Q
I2
--0
.04
44
4 E
I
10
Lin
earl
y di
stri
bute
d lo
ad
QI'
[c' [
c
QI2
[e
' [
M=
-Q
I [c
-~-
N=
-f [f-
Rl=
-Q
£
over
a p
ortio
n of
a s
pan
w=
18
0EI
T
7+
6/-
w'=
180
EI
T
7+
3 12
3
I
e2
x2]
x c
c2
x2]
_II (
x-d
]
II 3
(x-d
]
-3
--1
0-
-+
+
6--3
--3
0-
+
'.~'.
12
12
I I
12
12
c [e
'2
' e,
z
Q
x +
II 3
(x-e
)l]
+ IL
I5 (X
;~4]
c
c'
' 13
e,z
I e
~
, Q
le'
[ I
e']
--
Wo
= 1
80E
I x
R2
=-Q
1
-3
"/
.-.~.
N>-
-..I
[ e
e2]
x 7
+6
--3
-I
12
, Q
le'
w,=
-18
0EI
x
~ [
e e2
] x
8+9T
+3
[2
Con
L'd
§ B
eam
, lo
adin
g, a
nd
diag
ram
s of
mom
ents
A
ngle
s of
Su
ppor
t an
d sh
ear
forc
es
Ela
stic
cur
ve
rota
tion
Ben
ding
mom
ents
Sh
ear
forc
es
reac
tions
2.
C
antil
ever
Bea
ms
11
Con
cent
rate
d lo
ad
PI'
[Xl [3
X]
] w
'= ~:
[f [1-
11]]
at t
he e
nd
w=
3E
I 12
"2-
2i
M=
P(/
-x)
N=
-P
R=
-P
R
P P
I' w~
=O
Mm
v. =
PI
M=
PI
rl,'m
L.!.
w_
= 3
EI
for
x =1
P/
2 fo
r x
=0
rr--M
t---t
w,=
2E
I
~
_._
.-
12
Con
cent
rate
d m
omen
t M
x2
, M
x at
the
end
w
=
2EI
W =
Ei
M=
Mm
.. =
M
N=
O
R=
O
~
Mil
w~=O
M
=M
w
_=
2EI
'y
t ~
for
x=1
MI
w,=
EI
,M
c._
._.:
J
13
Uni
form
ly d
istr
ibut
ed
QI'
Xl [
X
w'=
QI2
E
x
M=
~I [I
-ff N
=-Q
[I-f]
load
ove
r th
e en
tire
span
w
= 2
4EI
12 6
-47
6EI
I R
=-Q
~Q
X2]
[
x x2
] Q
I +
-x
3-3
-+
-M
mu
.=T
12
I
12
1IL
1Il!l!!
Q.!
QI'
M
= Q
I Y
M
t
w~=O
fo
r x
=0
~
wm
.. =
8E
I 2
for
x=l
' Q
I2
...
--
-_
... -
w,=
S;E
I --_
._
--
Con
l'd
Bea
m,
load
ing,
and
di
agra
ms
of m
omen
lS
Ang
les
of
Supp
ort
and
shea
r fo
rces
E
last
ic c
urve
ro
tatio
n B
endi
11Jt
mom
enlS
Sh
ear
forc
es
reac
tions
14
Uni
form
ly d
istr
ibut
ed l
oad
QI'
[[
e lx'
w'=
QI'
[[I+~l
=:-Q
I [
C
x N=_Q[I_ILx~c]
R=
-Q
w=
--6
1+
---
M=
T 1
+/-
2/+
ov
er a
por
tion
of a
spa
n 24
E1
1 I'
2£
1 I
I
-4
£'
/I (X
-C)4
] _
£.+
/1 (x
-d]
II (X
-C)2
]
~
13 +
e c'1
3 12
e
3c'1
2 +
C
c'
l
'I'll,
x
QI'
[ e
en 1
w~=O
M
= Q
(c+
l)
C
Cl
-
W,=
24E
1 2+
6 T+
1'
2 ,
t fo
r x=
1 r-
----
.".N
~
Q/c
' [
c 1
• Q
I' [
e e'l
M
=
Q(c
+l)
w
'=4
El
1+
3/
w,=
-1
+-+
-6E
l 1
I'
max
2
for
x=c
' Q
lc
We=
2E
I fo
r x=
O
IS
Lin
earl
y di
stri
bute
d lo
ad
QI'
x' [
x w
=---
10-1
0-+
60
EI
I'
1 ,
QI'
x [
x w
=
12E1
T 4
-6T
+
M=
tQI[
I-fr
N
=-Q
[1-f1
' R
=-Q
m~~
x2
x3]
x2 x3
] +
5r
- r
+4r
- r
QI'
w~=O
Q
I M
= Q
I y
t M
max
= """
3 fo
rx=
O
~
w
=--
-15
EI
2 fo
r x=
1 w;=~ fo
rx=
1 12
£1
Con
cent
tate
d m
omen
lS
W=~
[3[=:1
'-w
'= ~;
[+-[T
n PI
[
x]
N=
-P
16 M
=T
1
-2/
R=
-P
at th
e en
ds·
I2E
I I
-2[fr
J 1;
£-,;4
' wo=
O f
or x
=O
; w~=w;=O
M=
PI
for
x=O
M.=-
~= P
I 2
2 PI
' PI
2 M
=-P
I fo
rx=
1
~
W,=
12E
1 fo
r x=
1 W
max
= 8
EI
---
-1
2 M
I
for
x="2
i
~-
--
• T
he r
otat
ion
angl
e at
the
rig
ht e
nd i
s ze
ro
~
o
2
Bea
m, l
oadi
ng, a
nd
diag
mm
s of
mom
ents
an
d sh
ear
forc
es
Con
cenb
aled
forc
e in
the
span
~~
Two
sym
met
tic
conc
enba
led
forc
es
~
:t ~"'I
c'
c \
TA
BL
ES
OF
DE
FL
Ecn
ON
S O
F SO
ME
ST
AT
ICA
lLY
IN
DE
TE
RM
INA
TE
BE
AM
S (B
EA
MS
CL
AM
PED
AT
BO
TH
EN
DS)
Ela
stic
cur
ve
w=
PB
£:£
'[3'
£-6
12
12
1
_ 3c
+c'
.!]+
1 1
+11/:
[x~c r
PB
C'
c<3
w'=
T/'
3r
for
x=c
PB
x' [ .. ,
xl
"--
-;r 3
---
+ 6
/ /'
/
II PB
[.!:
:E.]'
+
• 6
1 +
+II
.PB
[x-c
']'
• 6
1
PB
c'
2c'
-c
w'=
7"j3
-I-
for
x=c
Ben
diru
! m
omen
ts
cn [.£
M=
PT
/
3c+
c' .!]
+
-~I-I
+ILp<
x-c)
Shea
r for
ces
N=
-P c
',
3c+
c' +
11,1
' I'
1
M=
ph
' -x]
+ I N
=-p+
ltp+lt
.p
+ILp<
x-c)+
+ 1I •
• p<
x-c'
)
S~moments
II, =
cc'
, P
/'
c2c'
H
z=-T
P
II, =
cc'
P
1 cc'
Hz=
-TP
Supp
ort
reac
tions
cn
RI =
-r(3
c+
c')
P
c2
R2=
-/'3
(c+
3c')
P
R1
=-P
R2=
-P
Not
es:
I) T
he o
rigin
is
at t
he l
eft
end.
2)
In
the
expr
essi
ons
for
the
elas
tic c
urve
s th
e RO
l8tio
n B
= ~ is
use
d.
3) T
he s
uppo
n m
omen
ts a
re p
ositi
ve i
f
they
are
dir
ecle
d co
unte
rclo
ckw
ise.
4)
Ten
us a
fter
the
sign
I L refe
r onl
y to
cro
ss-s
ectio
ns x
> c.
~ ..
3
Bea
m,
load
ing,
and
di
agra
ms
of m
omen
ts
and
shea
r fo
rces
Uni
form
ly d
istr
ibut
ed l
oad
over
the
ent
ire
cros
s-se
ctio
n
II S
~Q
ml~t.
4 I
Uni
form
ly d
istr
ibut
ed l
oad
over
a p
orti
on o
f th
e sp
an
on t
he l
ift
5
m@~
c-+-
--c
, t
Sam
e, w
ith t
he l
oad
on
the
righ
t
~~2
Ela
stic
cur
ve
w=
QB
~ [1
-2~+~]
24
I'
I
I'
__
I_Q
B
wm
u -
384 1
for x=
2
W=.p!.
..{~ [
'='X-
2[2-
24
I'
/
c1
el
] x
c' [
-2
-+
-c-+
-6
-I'
I'
I
I
c cZ
]] -8
-+
3-
-1
IZ
_II (X
-C)4
} c
13
w=
qB {~
[~ [2£
-24
I
I 14
c'
c ]
-4
-+
4--2
+1
-I'
I
c1
c]
C4
] -6
-+
8--3
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Index
Absolutely stiff plates, 305 Adiabatic modulus of elasticity, 159 Adiabatic process, 156 Airy, G. B., 69 Airy (stress) function, 69 Albrey, T., 150 All-round compression (see Hydrostatic pres-
sure) Apparent modulus, 153, 154 Argyris, J. H., 12 Aron, H., II Axisymmetric deformation, 91
Backward differences, 355 Bars, 56, 170
See also Beams Beams, 169
See also Bending of beams Beltrami, E., 47 Beltrami-Michell's equations, 47, 161 Beltrami's strength theory, 63 Bending of beams, 169
application of Castigliano's theorem, 219 application of Rayleigh-Ritz and Bubnov-
Galerkin methods, 226-234 application of theorem of least work, 223 beams on elastic foundation, 190, 191 bending moment and shearing force, 173 boundary conditions, 179 combined action of lateral and axial loads,
198 differential equations of bending, 179 effect of shear (beams on elastic founda-
tion), 194 elastoplastic bending, 130 large deflections (elastica), 264 nonlinear bending, 201 reference tables of deflections of beams,
186, 403 shearing stress and shear deformation, 187
supports, 170-172 thermal bending, 209
Bending of frames, 243 complex frames, 248 simplest frames, 243
Bending of plates, 256 circular plates
axisymmetric bending of absolutely rigid plates, 317
large deflections under uniform load, 312 von Kiinmin's equations, 308
classification of plates from the viewpoint of their analysis, 305
Kirchhoff - Love hypotheses, 257 - 258 rectangular plates
application of Rayleigh - Ritz and Bub-nov-Galerkin method, 285
application of superposition method, 283 boundary conditions, 265 cylindrical bending, 264 differential equation of bending, 262 large deflections, 292 large deflections under uniform load, 298 M. Levy solution, 272 "membrane" stresses, 292 Navier's solution, 268 plate clamped along two opposite edges,
276 plate under distributed edge moments,
279 pure bending, 57 spherical bending, 265 von Kiinmin's equations, 298
thin and thick plates, 256 Bernoulli, J., II Bessel, F. W., 349 Bessel equation, 349 Bessel function, 349 Betti, F., 225 Biharmonic equation, 70
413
414 INDEX
Biharmonic operator, 262, 308, 311 Bimaterial assemblies, 81 Bland, D. R., 150 Body forces, 22, 25, 91 Boltzmann, L., 12 Boundary conditions
in bending of beams, 179 in bending of plates, 265 in elasticity theory, 23, 162, 164 geometric (kinematic), 214, 268 mixed,268 static, 23, 268 in torsion, 100, 101 in two-dimensional problems, 70
Bowing (see Bending) Bredt, R., 109 Bredt's theorem in torsion, 109, 164 Bryan, G. H., 340, 346 Bubnov, I. G., 279 Bubnov-Galerkin method, 227, 233, 285,
287, 298, 313, 351 Buckling of beams (bars), 324
bars on elastic foundation, 327 curved bars, 335 effect of initial curvature, 329 Euler's formulas, 324 large deflections (elastica), 331
Buckling of plates, 340 circular plates, 348 rectangular plates
plate compressed in two opposite directions, 346
plate simply supported along opposite edges, 344
plate uniformly compressed in one direction, 340
Bulk modulus, 43, 126, 161
Canonical equations, 229 Canonical form of stress surface, 29 Cantilever, 52, 71, 172, 173, 188, 231 Castigliano, A., 219 Castigliano's theorem, 219-221 Cauchy, A. L., 11, 29, 35 Cauchy's formulas, 35, 45, 50, 53, 66 Central differences, 355 Circular cylinder, 92 Circular plate
bending, 308-317 buckling, 348 See also Bending of plates and Buckling of
plates Circular ring, 86, 89, 92, 317 Circular shaft, 53, 163 Circulation of a vector, 109 Circulation of the shearing stress, 109
Claugh, R. W., 234 Cleavage plane, 121 Coating, 97 Coaxial cylinders, 94 Coefficient of compliance, 79, 80, 81 Coefficient of thermal expansion, 48, 94, 145 Collocation method, 358 Compatibility conditions, 35, 36, 67 Complementarity theorem, 26 Components of strain, 33, 37 Components of stress, 19, 20, 21 Compression, 20, 55 Concentrated forces and moments, 170 Conditions of compatibility (see Compatibility
conditions) Constants, elastic (see Hooke's law) Coordinate functions, 214 Coulomb, C. A., 11, 62 Courant, R., 12 Creep, 144, 146, 147, 154 Curved bar, 334 Cylinder
symmetrical deformation, 94 thermal stresses, 94 thick, under pressure, 93
Darrow, K. K., 3 Deflection of beams (see Bending of beams) Deflection of foundations, 254 Deformation method, 235, 367 Dilatation, 38, 126, 161 Direction cosines, 22, 23, 26 Displacement method, 235, 367 Displacements, 33 Drucker, D. C., 12 Duhamel, J. M. C., 44, 47, 48, 161, 162 Duhamel-Neumann's equations, 47, 48, 162 Duhamel-Neumann's hypothesis, 44, 161
Elastic limit, 42 Elasticity theory
direct and inverse problems in, 45, 161 elementary problems in, 49, 163 hypotheses in, 15, 16 semi-inverse method in, 46, 161
Ellipsoid, stress, 29, 30, 55 Elliptic bar, 104, 105, 165 Energy
functional, 227, 286 law of conservation of, 215 strain, 60, 228, 235, 239
Enthalpy, 156 Entropy, 155 Equilibrium conditions, 22, 25, 26, 67, 160,
176 Euler, L., 11
Euler's force, 326 Euler formulas, 191
Filon, L. N. G., 74 Filon's solution, 74, 164 Finite differences, 353 Finite-element method, 6, 7, 360, 372 First-order method, 355 Aexible plates of small deflection, 306 Aexural rigidity of plates, 59, 256 Force method, 254 Forward differences, 354 Fourier, 1. B., 73, 269, 341 Fourier series, 73, 269, 341 Four-point bending, 174 Fracture, 115 Fracture mechanics, 121 Fracture modes, 117, 118 Fracture tests, 121 Fracture toughness, 121 Frames, 243
See also Bending of frames Free energy, 156 Free enthalpy, 156
Galerkin, B. G., 5 Galilei, G., 62 Generalized coordinates, 214 Generalized displacements, 214 Generalized forces, 215 Generalized Prandtl' s formula, 104 Germain, S., 11, 256, 262 Gibbs, 1. W., 156 Gibbs' potential, 156 Gilbert, D., 5 Grave, D., I Green, G., 103, 109 Green's formula, 103, 109 Griffith, A. A., 115 Griffith theory, 115 - 117, 165 Gross, B., 150
Hardening, 42 Harmonic function, 47 Harmonic operator, 47, 69, 263 Hencky, H., 12 Hermit functions (polynomials), 365 Hill, R., 12 Hollow shaft, torsion of, 110 Homogeneous deformation, 16 Hooke, R., 11 Hooke's law, 16, 32, 38-44, 45, 53, 67, 91,
161, 239 with consideration of thermoelastic strains,
43,44 Huber, M. T., 64
INDEX 415
Huber's theory, 64 Huber-von Mises-Hencky theory, 64, 65,
126, 163, 164 Hydrostatic pressure, 31, 32, 55, 125, 138 Hypothesis of plane cross sections, 175
Ideal plasticity, 132 Ilyushin, A. A., 128 Ilyushin's theorem on unloading, 130 Influence coefficients, 251 Interpolation polynomials, 354 Invariants of stress (See Stress invariants) Irwin, G. R., 117, 165 Irwin's stress intensity approach, 117 - 119,
165 Isochrometric lines, 384, 385 Isoclinic lines, 384 Isothermal process, 156 Isotropy, 16
Karman, T. von, 292, 308 Kelsey, S., 234 Kelvin, Lord (W. Thomson), 4, 12 Kirchhoff, G. R., 11, 256 Klapeiron B. P. E., 219 Klapeiron's theorem, 219 Kron, G., 12
Lagrange, 1. L., 11, 262 Lagrange theorem, 216, 217, 227 Lame, G., 30, 39, 47 Lame constants, 39, 40 Lame equations, 47, 48, 162 Lame problem, 93 Laplace, P. S., 47, 309 Laplacean operator, 47, 309 Lateral force, 173
(See also Shearing force) Least work, principle of, 222 Leibnitz, G. W., 62 Levy, M., 256 Levy theorem, 68, 85, 164-Light, polarized (see Polarized light) Linear viscoelasticity theory, 150 Lode, W., 12 Love, A., E. H., 11
Mariotte, F., 62 Martin, H. C., 234 Maxwell, 1. C., 12 Maxwell model, 149 Maxwell's theorem of reciprocity of dis-
placements, 225 Membrane forces, 292, 311 Membranes, 256, 306 Mesh method, 353
416 INDEX
Method of initial parameters, 183 Michell, A. O. M., 47 Mises, R. von, 12 Mohr, 0., 62 Mohr circles, 62, 63 Mohr strength theory, 62, 65 Moire method, 387 Moment of inertia, 176, 210 Multiply connected region, 37, 103, 112,
113, 164 Muskhelishvili, N., 91
Nadai, A., 12, 31, 65 Nadai hypothesis, 65 Navier, H., 11,257 Navier's solution, 268-272 Neutral axis (surface), 176 Newton's interpolation formula, 354 Nonlinearity, 16
Orawan, E., 117 Orthogonal functions, 76
Photoelasticity, 69, 380 Plane strain, 66, 164 Plane stress, 66, 164 Plastic deformation, 16, 125 Plastic flow, 124, 125, 138, 165 Plasticity, 123 Plastics, 143 Plates (see Bending of plates and Buckling of
plates) Point-slope formula, 355 Poisson, S. D., II Poisson's equation, 100 Poisson's ratio, 40, 43, 161 Polar coordinates, 82, 164 Polariscope, 380, 381 Polarized light, 381 Polarizer, 380 Polymers, 143 Polynomials, 70, 164 Prager, W., 12 Prandtl, L., 101 Prandtl-Reuss equations, 141 Prandtl's formula, 101, 104 Prandtl's membrane analogy, 375 Prandtl's stress function, 100, 164 Prerupture deformation of plastics, 149 Pressure, 92 Principal axes, 26 Principal directions, 26, 37 Principal plane, 27 Principal strain, 37 Principal stress, 27, 30
Principle of least work, 222 of Saint-Venant, 46 of superposition, 16 of virtual displacements, 215, 216
Prismatic bar, 49, 164 Proportional limit, 41 Pure bending, 49, 57
Rayleigh, Lord (J. W. Strutt), 226 Rayleigh - Betti theorem, 225 Rayleigh-Ritz method, 226, 229, 286 Recurrence formulas, 356 Reciprocal property of displacements, 224 Reciprocal theorem for forces, 217 Reciprocal theorem for works, 225 Rectangular plate (see Bending of plates and
Buckling of plates) Redundant unknowns, 172 Relaxation function, 151 Relaxation of stress, 144, 146 Relaxation time, 150, 152 Research models, I Residual deformations, 15 Retardation function, 151 Retardation time, 151 Ribiere solution, 73, 164 Rigid-body displacements, 35 Ring (see Circular ring) Ritz, W., 226 Ros, M., 12 Rotation, 33 Ruge, A., 375 Runge-Kutta method, 356 Rupture, 148
Saint-Venant, B. de, II, 36, 306 Saint-Venant-Uvy-von Mises equations,
141 Saint-Venant's conditions of compatibility,
36,67 Saint-Venant's hypothesis, 98 Saint-Venant's principle, 46, 161 Saint-Venant's semi-inverse method, 46 Saint-Venant's torsion function, 98, 164 Schleicher, F., 12 Second-order methods, 356 Section modulus, 72, 73, 106, 130 Semi-infinite beam on elastic foundation, 193 Shear modulus, 41, 43, 126, 161 Simmons, E. E., 375 Simple loading, 142 Simply connected region, 37, 103, 161 Small elastoplastic strains, 125 Solution procedures, 45 Specific heat, 157
Spherical tensor, 31, 160 Spring constant, 191 Statically indeterminate problems (systems),
26, 160, 172 Stiffness matrix, 240, 363, 368 Strain
at point, 33 compatibility of, 35, 36, 67, 161 components of, 33, 37 effective, 127 invariants, 37, 161 nominal, 124 normal and shearing, 33 plane, 27 principal axes of, 37, 160 principal planes of, 37, 160 rotation, 33 surface, 37 tensor, 37, 161 thermoelastic, 43 "true," 124
Strain energy, 60, 61, 163 Strain-hardening coefficient, 124 Strain gauges, 375 Strain rosette, 379 Strength
coefficient, 124 theories (hypotheses), 60-65, 163 ultimate, 42
Stress at point, 19 components of, 19, 86, 160 concentration, 164 deviatoric, 31, 32, 160 effective, 126 invariants of, 26, 29, 32, 126, 160 mean, 18 nominal, 123 normal and shearing, 20, 21 octahedral, 31, 126, 160 plane, 27 principal, 26, 31, 160 radial, 82-85 relaxation, 144, 146 shearing, 20, 26, 30, 82-85, 112, 160,
178, 179 signs of, 20 surface, 29 tensor, 20, 21, 28, 31, 39, 160 "true," 123
Stress ellipsoid, 30, 160 Stress functions, 69, 100, 164 Stress-strain relation, 42, 124, 156, 158, 161
See also Hooke's law Stress surface, 29 Structural analysis, 8-10
INDEX 417
Superposition, principle of, 16, 162, 279, 283 Synge, J. L., 12
Temperature, stress due to nonuniform dis-tribution of, 209
Tension, 20, 56 Thermal expansion, 48, 94, 145 Thermal stress, 94, 209 Thermodynamic potential, 155 Thermodynamics of viscoelastic deformation,
155 Thermoplastic materials, 143 Thermosetting materials, 143 Three-point bending, 174 Time effects in plastics, 144-148 Timoshenko, S. P., 279 Topp, J. L., 234 Torque, 55 Torsion
boundary conditions in, 54 circular shaft in, 53 displacements in, 53, 54 elliptic shaft in, 104 multiconnected sections, 112 of prismatic bars, 53 of rectangular shaft, 107 section modulus in, 106 stress function for, 98 of thin-walled sections, 100, 108, 110
Torsional rigidity, 55, 112, 165 Trapezoidal formula, 356 Tubes
subjected to internal and external pressure, 93
torsion of (see Torsion) Turner, M. J., 234 Twist angle, 55, 109, 112, 114
Ultimate axial force, 130 Ultimate bending moments, 130
Variational methods, 213 Virtual displacements, 215, 216 Viscoelasticity, 143 Viscoelastic deformation, 155 Voight, W., 12 Voight-Kelvin model, 149
Wheatstone bridge, 377 Winkler, E., 190 Winkler's hypothesis, 190
Yield stress, 42, 164 Young, T., II Young's modulus, 40, 41, 145, 161
