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Some methods for structural analysis
S o m e m e t h o d s for structural analysis J T S Wang & N F Wang"
Abstract Solutions based on fundamental aspects of beam theory are used for establishing some new methods in the structural analysis of continuous beams and frames. Hew slope.deflection equations are given which represent an improvement over the conventional slope.deflection method. A method using end moments and deflections as prima~ quantities is presented as an alternative procedure to the slope.deflection method. This alternative method is referred to as the moment - or curvature. deflection method. The method and some results based on the alternative procedure are compared with the slope.deflection method throtigh illustrative examples. A third procedure of using Fourier series in conjunction with the Stokes' transformation is discussed. The Stokes' transformation procedure suggests that the choice of end curvatures and deflections as primary quantities used in the moment.deflection method is more natural than choosing the end slopes and deflections.
Introduction The moment distribution and slope-deflection methods are two basic conventional methods commonly used in the structural analysis of continuous beams and rigid frames. Deflection, slope, bending moment and shear force are the four unknown quantities at each section of a beam under bending. Both of these methods express shear and moment in terms of rotation and deflection at two ends of each beam or a segment of • a beam, and the problems for analysis are formulated by satisfying equilibrium conditions at junctions of adjacent segments. While the slope.deflection method solves directly the equations established from equilibrium conditions, the moment distribution method follows an iterative procedure. Detailed discussions on these methods can be found in textbooks on structural analysis 13. The original ideas behind these methods may be found in References 4-7. For the moment distribution method, Gere 8 gives the most extensive treatment. Wang and Salman 3 give brief and concise presentations on the historical developments of these methods. The slope-deflection method in matrix format for analysing statically indeterminate structures has become more emphasised as computers are increasingly used. While the slope-deflection method is well established and widely used, derivation of the basic equations based on the superposition principle is cumbersome. Furthermore, the indeterminate fixed end moments involved must be determined using a separate method. These shortcomings may be removed by considering directly the general solution for the bending of beams from the basic structural mechanics points of view. Also, in some cases, the use of quantities other than slope and deflection as primary unknowns at each end may become more desirable. These are the reasons which motivated this study. In what follows, explicit expressions for a new set of slope and deflection equations will be presented first. As an alternative procedure, the bending moment and deflection at each end of a beam or a segment of equations established by requiring continuity in rotation at each joint in conjunction with force equilibrium conditions is solved for the unknown end
moments and deflections. The method may be called the moment-deflection method. Since the moment is directly proportional to the curvature of the deformed beam, the method may also be referred to as the curvature-deflection method. Although the slopedeflection method is well established and widely used, the present moment-deflection method is comparable to the slope-deflection method in concept and at times may be easier to apply. In the conventional slopedeflection method, determination of fixed end moments requires an analysis of an indeterminate problem itself. Consequently, only simple loading conditions are routinely used. The present method, however, provides direct solution without relying on other supplementary procedures. Furthermore, the moment.deflection method may be applicable to structures having semi. rigid joints. w
I
.~F----T
M~.~V+dV
P
w~ X
0
- ~ d xl~-
b
Figure I Geometry and =ign convention
Basic equations The deflection, w, of a beam under a loading function p(x) is governed by the following well known equation -Elw"
=p
(1)
in which a prime denotes differentiation with respect to x, which is the coordinate along the longitudinal axis. The shear V and bending moment M are V --- E l w "
(2)
M = Elw"
(3)
respectively. The geometry and sign convention are shown in Figure 1. The general solution of equation (1) is Elw = A w 3 + Bx 2 + Cx + D + P(x)
(4)
in which "School of Civil Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
140
0950 - 0618/90/030140 - 07 © 1990 Butterworth-Heinernann Ltd
P(x) = I~ p(x)dxdxdxdx
(5)
3 CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
The loading p, which may involve singularity functions, is considered to be explicitly integrable. Otherwise, certain series representations may be used. The integration constants A, B C and D may be expressed in terms of either end slopes and deflections, or end • moments and deflections.
New slope-deflection equations By choosing slope, 0 = w ', and deflection, to, at ends a and b of a beam segment of length L as the independent quantities, i.e., w(0)=w,, w(L)=%,
w'(O)=O,, w ' ( L ) = O b
(6)
than the conventional slope-deflection equations. As an example for illustrative purposes, the function P(x) for the following class of loading p = -q(x/L)"
(15)
P(x) = - [n!/(n + 4)! ]qx"+4/L"
(16)
is
As a result P(O) = P'(O) = P " (0) = 0
(17)
P(L) = - [ n!/(n + 4)!]qL'
(18)
P ' ( L ) = - [n!/(n + 3)[]qL 3
(19)
P " ( L ) = - [n!l(n + 2)!]qL 2
(20)
the integration constants in equation (4) are found to be A = El(Oh -- O,)/L 2 - 2EI(wb - w , ) / L 3 -
+P'(O) ] IL 2 - 2 [P(O) - p ( L ) ] / L a
[P'(L) (7)
B = - E I ( 2 0 , + Ob)/L -- 3 E I ( % - w , ) / L 2 + [2P'(O)
+ P'(L)]/L + 3 [ / ' ( O ) - P(L)]/L 2
(8)
- e'(0)
(9)
D = E l w , - P(O)
(10)
C = E/O,
P'(L)
= - [1/(n + l ) ] q L
(21)
By substituting equations (17) to (21) into equations (11) and (12) for the fixed end conditions with w. = wb = 0, = #b = O, the fixed end moments are found to be ME° = 2 P ' ( L ) / L - 6 P ( L ) / L 2
= - [2(n + 1)!/(n + 4)!]qL 2 The end m o m e n t s M, and bob at x = 0 andL respectively from equation (3) b e c o m e the new slopedeflection equations as follows M , = - 4 E I O , IL - 2EIOblL + 6El(wb -- wo)lL 2 + MF,
(22)
MR, = P " ( L ) - 4 P ' ( L ) / L + 6 P ( L ) / L 2
= - [(n + 2)!/(n + 4)!]qL 2 Table I
(23)
Fixed end moments for v a r i o u s values o f n
(11) Mb ffi 4EIOb/L + 2EIO,/L - 6 E 1 ( % - w , ) / L 2 + M R ,
0
1
2
10
1/12 1 / 12
1/30 1/20
1/60 1/30
1/1092 11182
(12) - MF° I(clL 2) - MFb I(oL 2)
in which MF° = P " ( O )
+ 2 [2P'(0) + P'(L)]/L
+ 6 [P(0) - P(L)]/L 2
(13)
MR, = P " ( L ) - 2 [2P'(L) + P ' ( O ) ] / L
- 6 [P(O) - P(L)]/L2
(14)
The luted end moments corresponding to various values of n are given in Table 1, where the results for n = 0 and 1, corresponding to uniform and linearly varying loadings, are seen to agree with the well known exact solutions. For sinusoidal loading p = q s i n ( m x x / L ) , t h e function P(x) is give by P(x) = q L ' / ( m x ) 4 s i n ( m w x / L )
Except for the sign conventions for 0,, 0b, W, and wb which may differ from author to author, equations (11) from which and (12) are in the usual form of slope-deflection P(O) = P ( L ) = P"(O) = P"(L) = 0 equations, in the conventional slope-deflection method, a separate analysis method is needed to solve the P'(O) = p m (0) = qL3/(m'lr) 3 statically indeterminate problem for the Fu
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
(24)
(25) (26) (27)
(28) (29)
141
If the loading function is represented by a Fourier sine series, then the summation for m ranging from 1 to Qo should be made. For a beam under a concentrated load F at x = and a moment M at x = ~ the loading function expressed in terms of a singularity function x - ot n is
E/O b = El(Wb -- wa)/L +
(2Mb + Ma)L/6 + P'(L)
- [2P"(L) + P"(O)]U6 -
[P(L)
-
P(O)]/L (44)
V= = (M~ - Mb)/L + [P"(L) - P"(O)]/L
-
Pro(0) (45)
p = F(x-
~) -j + M ( x -
(30)
vl>-2
Vb = ( M = -
Mb)/L + [P"(L) - P"(O)]/L - P#'(L) (46)
l:or which P(x) = F ( x - ~)3/6 + M ( x - ~1) 212
(31)
P(O) =P'(O) =e"(0) = 0
(32)
P(L) =F(L - ~)3/6 + M ( L - ~)2
(33)
P'(L) = F(L
~)2 + M(L - 77)
(34)
~) + M
(35)
P"(L)
-
=F(L-
At each joint, the end m o m e n t s of adjacents m e m b e r s are related through the m o m e n t equilbrium condition. By requiring continuity in rotation at each joint together with additional force equilibrium equations, one then establishes a system of simultaneous algebraic equations for the unknown m o m e n t s and deflections. We refer to this procedure as the moment-deflection method. The m e t h o d may best be illustrated by examples.
Examples
The fixed end m o m e n t s b e c o m e MF, = k(1 - k)2FL + 2(1 - s)M - 3(1 - s)2M
(36)
Mro = E(1 - k)FL + M - 4(1 - s)M + 3(I - s)2M (37) in which k = ~/L and s = ~I/L. Based on equations (11) and (1 2) as the new slopedeflection equations, the procedure for structural analysis follows that of the conventional slope.deflection method.
In order to illustrate the moment.deflection method and to make comparisons with the slope-deflection method, the following examples given in the textbook by Wang and Salman 3 for illustrating the application of the slope-deflection method to statically indeterminate beams and frames are considered. It should be noted that the sign convention used in the present method of analysis is different to that used in Reference 3. E x a m p l e 9.3.1 o f Reference 3: A n a l y s e the t w o s p a n continuous b e a m s h o w n in Figure 2. q= 2 k i p s / f t
Moment-deflection equaUons and procedure By choosing M and w at ends a and b of a beam segment of length L as the primary quantities, i.e.,
h~rrB
Lt = 2 0 f t -
[=
constant EZ L z = 30 ft
:i--
:'~
w(O) = w,, w(L) = % , Elw"(O) = M , ,
(38)
EIw"(L) = Mb
the integration constants A, B C and D in equation (4) are found as follows A = {(Mb -- M a ) / L - [P"(L) - P"(O)]/LI/6 B = [M=-
-
Figure 2
(39)
(4O)
P"(0)]/2
C = E1(Wb - w~)/L
25
t
Example 9.3.1
F = 2 4 kips
,
,
Iq, = 0 . 9 k i p s / ~ , %
A 21 ~1~_ kL_~,.~,=~_i0 f t ]= Ll:16ft
(Mb + 2M~)L/6
4
qz = Z k i p s / f t
÷
= ==
- [P(L) - P(O)]/L + [P"(L) + 2P"(O)]L/6
,
(41) D = EIw= - P(O)
(42)
The rotation and transverse shear force at a and B expressed in terms of w,, wb, Mo,/v/b and the loading are considered as the moment-deflection equations ElOa = El(wb -- wa)/L
-
(2Ma
-
P(O)
]/L
(43)
142
2 3 Figure 3
4 5
Example 9.3.2
.Solution. Since there is no m o m e n t at A and C, and no
deflection at A, B and C, only one condition, 02 = 03, needs to be satisfied. Using the moment.deflection equations, one obtains
+ Mb)L/6 + P'(O)
+ [ P" (L) + 2P" (0) ] L/6 - [ P(L)
5] Lz= 2 4 f t
EI02 = M2LI/3 + qL~/24
(47a)
El03 = - M ~ . 2 / 3 - qI.~124
(47b)
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
Equating equations (15) and (16) and knowing that h/2 = M~ because of moment equilibrium at the joint B, one obtains for q = 2 kips/ft, L~ = 20 ft and L 2 = 30 ft,
= -(q/S)(L~ = - 175 ft-kips
M2 =
M~
+
I.~)I(L,
+ L2)
(48)
which agrees exactly with the solution given in Reference 3. For this example, only one equation is used to solve the problem while three simultaneous equations for 0^, 0B and 0c are involved when the slope. deflection m e t h o d is used. Clearly, if ends A and C are fixed, then the present method would involve three equations and only one equation would be involved for the slope-deflection method.
Example 9.3.2. of Reference 3: Analyse the continuous beam shown in Figure 3. Solution: Using the numbering in Figure 3 and apply-
ing the moment.deflection equations
slope.deflection method involve two simultaneous equations.
Example 9.4. I of Reference 3: Compute the midspan deflection of a simply supported uniformly loaded beam as shown in Figure 4 utilising a joint at the midspan. Solution. Using the numbering in Figure 4 and apply-
ing the moment-deflection equations
(49)
El, 02 = (2M2 + M,)L,/6 - ql L3/24 -FL2k(1 - k2)/6
(56)
EI03 = -2EIw2/L - M3L/6 - qL3/192
(57)
V2 = -2M2/L + qLI4
(58)
V3 = 2M3/L - qL/4
(59)
By knowing that /4~ = /43 from the m o m e n t equilibrium and w2 = w3 = we, the continuity in rota. tion 02 = 03 and the force equilibrium V2 - V3 = 0 at joint B lead to the following two equations
EllOi = -(2MI + M2)Lx/6 + qlL~/24 + Ffk(1 - k)(2 - k)/6
E102 = 2EIw2/L + M2L/6 + qL3/192
4EIwBIL + M2L/3 = -qL3196
(60)
M2 = qL2/8
(61)
from which one obtains the well known result
(50) (62)
EIwB = - (5/384)qL 4 E1203 = -(2M3 + M4)/.,zl6 + q2/-~/24
(51)
From the moment equilibrium at joints A, B and C and by knowing that/45 = -q2L~z/2, one has M2 =M3 and M4 -- M5 = - 3 6 ft-kips
(52)
The present method involves two equations for deter. mining we and M2 while the slope-deflection method would involve four equations for 0^, 0s, 0c and we. Clearly, if the beam is fixed at both ends, then the present method would involve for equations while the slopedeflection method would involve only two.
q kips/ft
r~,~ L"=Lz-I ~ F =40 kips B
A l"
B L
I
C ,1 I
I
23 B
Figure 4
q
2
Lz
for q2 = 2 kips/ft and L 3 = 6 ft. With q~ = 0.9 kips/ft, F = 24 kips, k = 3•8, Ll = 16 ft, L2 = 24 ft, II = 21, and/2 = Is = 51, the two required conditions 0~ = 0 and 02 = 03 lead to the following two equations
M~ + 3.2M2 = - 3 3 2 . 5 5 ft-kips
L~
4
Example 9.4.1
2M~ + M2 = - 2 0 3 . 8 5 ft-kips
3
C
51
Figure 5
Example 11.7.3
14
I
Example II. 7.3 of Reference 3: Analyse the rigid frame having two hinged supports as shown in Figure 5.
(53)
Solution. The axial coordinate from A to B, B to C, and C to D is considered to be in the positive direction. Us(54) ing the numbering in Figure 5, the moment deflection equations give
Solving equations (22) and (23), one obtains
Ell 02 = Ell w2/Ll + M2LI/3 + qL~/24
(63)
I/2 = -M2/Lm + qL,/2
(64)
M l = - 5 9 . 2 1 7 ft-kips and M 2 = - 8 5 . 4 1 7 ft-kips
(55) which agree exactly with the results given in Reference 3. For this example, both the present method and the
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
E1203 = -(2M2 + M , ) ~ / 6 - FI~k(1 - k)(2 - k)/6
(65)
143
El204
-~"
(2M4 + M3)L2/6 + F ~ k ( 1 - k2)/6
EI305 = - E l a wsIL3
-
(66) (67)
MsL313
Vs = M5/L3
(68)
Clearly, M2 = M3, M4 = M5 and w2 = - w s . The conditions to be satisfied are V 2 + V 5 = O, r M 3 + E1302 = El303
and 04 = 05 (69)
in which the parameter r is introduced to account for the semi.rigidity of joint B. If the joint is rigid, r = 0. Using equations ( 6 3 ) - ( 6 8 ) with q = 0.5 kipsflt, L] =
L3 = 16 ft, L2 = 20 It, k = 0.4, 11 = 31, 12 ffi 51,/3 = 21 and r = O, the conditions given in equation (69) result in the following equations
handbooks, the use of the conventional slope-deflection method is normally limited to such simple loading conditions. While the new slope-defiection and momentdeflection methods are more direct and flexible than the conventional slope-deflection method, the loading function must be explicitly integrable. For cases when loading functions are not explicitly integrable, a method using Fourier series in conjunction with the Stokes' transformation is presented here. Dicussions on the Stokes' transformation used in the Fourier series analysis can be found in Reference 9. Representing any loading function Pm sinot,nX for 0 < x < L
p(x) = ~
(74)
m~l
and the displacement function oa
M3 - M4 = 64
(70)
- ( 9 / 1 6 ) E 1 2 w s + (9r + 140)M3 + 30M4
=
for
O
IV,
at
x = 0
Wb
at
x = L
-10496 (71)
w(x) =
(72)
(3/16)E12w5 + IOM3 + 60M4 = - 2 6 8 8
Solving equations ( 7 0 ) - ( 7 2 ) for r = 0 M3 = - 1 3 . 4 7 4 kips-ft, M4 = - 7 7 . 4 7 4 kips-ft
and
Elws = 2234.796 kips-ft 3
(73)
which agree with the results given in Reference 3. Since the frame is supported by hinges at A and 13, the present method involves fewer equations than the slopedeflection method. While the slope.deflection method cannot account for the semi-rigidity of the joints, it is a simple matter to include this effect in the present method. To show the effect of r at joint B, some numerical results are given in Table 2. It may be noted that r ranging from 10 to 0o corresponds to the joint varying from rigid to hinged connections respectively. It is seen in Table 2 that the moment at joint B reduced from the rigid joint case by about 50% when r = 40 and by over 90% when r = 400. Table 2
W~sinctmx
Effect of semi-rigidity of joint B (M 4 = M 5 = M 2 -
(75)
in which ¢=m= m x / L and Pm and Wm are Fourier coefficients for the loading and displacement functions respectively, the derivatives of the displacement function given in equation (75) following the Stokes' transformation are
w'(x) =
(Wb -
W,)IL+ ~ W'=cosot,x m=l
for 0 _< x < L
(76)
where W'm = t~, W., - (2/L) [ W. - ( - 1)mWb]
t~,~W" s m t ~ x
for 0 < x < L
64)
w"(x)= r
M 2 = M 3
r
M2 = M 3
0 10 20 30 40 50 60 70 80 90 1O0
- 13.474 - 10.894 - 9.143 - 7.877 -6.919 - 6.169 - 5.565 - 5.069 -4.655 - 4. 303 - 4.000
150 200 300 400 500 600 700 800 1,0OO 10,000 1O 0 , 0 0 0
- 2.960 - 2.349 - 1.662 - 1.286 - 1.049 - 0.886 -0.766 - 0.675 -0.546 - 0.057 - 0.0057
at
x -- 0
at
x =L
(77)
and ao
w " (x) = (Kb -- K=)/L + ~ ,
W~' cost~=x
m=|
for O - - x _ < L
(78)
where
Fourier a n a l y s i s with Stokes' transformation Since the fixed end moments corresponding to simple loading conditions are commonly given in textbooks or
144
W," = (2/L)ot2 [ W, - ( - ( 2 / L ) [x= - ( -
1)mWb] 1)%b] - cx~Wm
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
L. The beam is under uniform load q distributed trom x = 0 to k L for 0 k 1. The exact solution for the fixed end moment at x = 0 is
and
w'~(x) = - ~
t~,~WZ sinctmx
m=l
(79) M(O)
'
ln the above, Wo and W~ are deflections at ends a and b, and Ko and Kb are the end curvatures. By substituting equations (74) and (79) into the governing equation (1) for the beam deflection, and collecting like terms in the resulting equation, the unknown Fourier coefficients for displacement Wm are expressed in terms of the end curvatures K, and K~, deflections W° and W~ and the loading coefficients p,,, as follows
=
(87)
~qL=/8
in which /~ = I - ( 1 - k)212 - (I
-
k)2]
The corresponding solution for the fixed end moment /v/s at x = 0 based on the Stokes' transformation using the curvature.deflection method is found to be
Ms(O) =aqL2/8 W . = (p./n,~)/EI + (2/L)I~. [ W= - ( -
1)'Wb ]
-(2/L)/~m [~, - ( - 1)%b]
(80)
(89)
in which o=
ce = (48hr 4) ~ It is interesting to note that the general solution is obtained in terms of end curvatures and deflections, which indicates that the moment.deflection method presented earlier appears to be more natural than the conventional slope-deflection method. The slopes and shear forces at the end sections expressed in terms of Wo, Wb, Ko and K~ become 0. = (Wb -- I4,'=)//., -- (L/3)K= -- (L/6)Kb OD
(81)
(88)
[ 1 - cos(/o'nT) ]/m
(90)
m=|
By taking 60 terms in the series for
Ob
(Wb -- W,)/L + (L/6)~<. + , ( L / 3 ) K
b
/~ = k 3 [ 1 + (1 - k)3x2/(3k)
o0
+
(82)
(mffil
Letting Wo = Wb = 0, = Ob = 0, we obtain the following fixed end moments for a beam
+0r2/2)(1
M,. = EX,. = ( 2 / L )
[2 + (-l)m](p=/=~)
go
[I + 2 ( - 1 ) ' ] ( p ~ a ~ ) (84)
~ p,./CX,.
~_, (k/m3)sin(kmx)/(1 - k2m 2)
(94)
m ffi l
m=l
K b ) / L -F
(93)
(83) in which c~ = 2hr
MFo = E&b ffi - ( 2 / L )
(92)
The corresponding solution for the fixed end m o m e n t M. at x = 0 using the curvature.deflection equation with the Stokes' transformation is found to be
m==l
E/(Ka -
k)(1 - 2k/3) + (1 - k)]
Ms(0) = o~(3qL2/r 3)
m
Va =
--
(85)
It should be noted that when rn = 1/k, the term in equation (94) must be determined from a limiting process. For such cases when rn = 1/, the term is given by
l,~mI [ sin(km~)/( l - k2m2) ] = ~r/2
(95)
,'O=1
As a result, equation (95) b e c o m e s Vb -- E/(K a -
Kb)/L -F ~
(-- l)"*pm/ol.
(86)
m=l
ot = k 4 + 21~r
(k/m3)sin(km~r)/(1 - k2m 2)
(96)
m=l
The subsequent procedure for analysis will be the same as that of the curvature-deflection method. For illustrative purposes, two examples are presented. The first example considers a uniform beam of length L which is fixed at x = 0 and simply supported at x ffi
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
km# l By taking 60 terms in the series for ¢xin equations (96) for various values of k ranging from 03 to ], o~agrees with/3 given in equation (94) to the fourth digit.
145
Conclusions A new set of slope-deflection equations is presented that can be used more directly than the conventional slope. deflection equations. An alternative procedure referred to a s the moment- or curvature.deflection method, • which expresses joint rotation and shear force in terms of end moments and deflections, is discussed. In the moment.defiectin method, a system of equations is assembled by requiring continuity in rotation together
with equilibrium conditions. The method, similar in concept t o t h e s l o p e - d e f l e c t i o n m e t h o d , can i n c o r p o r a t e t h e effect of semi-rigidity of the joints. When the loading f u n c t i o n is n o t e x p l i c i t l y i n t r e g r a b l e , t h e m e t h o d b a s e d o n t h e F o u r i e r a n a l y s i s w i t h S t o k e s ' t r a n s f o r m a t i o n is presented, and the procedure follows that of the momentor curvature.deflection method.
References 1 Hsleh, Yuan-Yu. Elementary Theory of Structures, 2nd edn. Prentice Hall Inc., Englewood Cliffs, N J, 1982 2 Maugh, L C. Statically Indeterminate Structures, 2nd edn., Wiley,
blew York, 1964 3 W a n g , C K and 8 a l m a n , C G. Introductory Structural Analysis,
Prentice Halt Inc., Englewood Cliffs, blJ, 1984 4 Crolm, H. 'Continuity as a factor in reinforced concrete design', Prec., ACI 1929, 25, 669.708 5 Cross, H. 'Simplified rigid frame design, Report of Committee
301, Prec., ACI, 1929, 2 6 6 M e n d e r l L H. 'Die Berechnung der Sekundarspannungen, Welch im einfachen Fachwerke in Folge Starter Knotenverbindungen auftreten', Forster's Bauzeitung 1880, 4 5 , p,34 7 Mohr, O. 'Die Berechnung der Faehwerke mit Starren Knotenver. bindungen', Der Civiltngenleur 1892, 38, 577-594; 1893, 39,
67.70 8 Gere, J M. Moment Distribtuion, Van blostrand Rheinhold Com-
pany, Princeton, blJ, 1963 9 (:hung, H. 'Free vibration analysis of circular cylindrical shells,
J. Sound and Vibration 1981, 74 (3), 331-350 Notation
E I M p r V
w x
0
modulus of elasticity area moment of inertia bending moment transverse loading parameter for semi-rigid joint transverse shear force transverse displacement axial coordinate rotation
Acnowledgement This paper was originally presented in Engineering Structures Vol.12, July, 1990 and is reproduced here by kind permission.
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CONSTRUCTION & BUILDING MATERIALS Vol. 4 NO. 3 SEPTEMBER 1990
Abstract Solutions based on fundamental aspects of beam theory are used for establishing some new methods in the structural analysis of continuous beams and frames. Hew slope.deflection equations are given which represent an improvement over the conventional slope.deflection method. A method using end moments and deflections as prima~ quantities is presented as an alternative procedure to the slope.deflection method. This alternative method is referred to as the moment - or curvature. deflection method. The method and some results based on the alternative procedure are compared with the slope.deflection method throtigh illustrative examples. A third procedure of using Fourier series in conjunction with the Stokes' transformation is discussed. The Stokes' transformation procedure suggests that the choice of end curvatures and deflections as primary quantities used in the moment.deflection method is more natural than choosing the end slopes and deflections.
Introduction The moment distribution and slope-deflection methods are two basic conventional methods commonly used in the structural analysis of continuous beams and rigid frames. Deflection, slope, bending moment and shear force are the four unknown quantities at each section of a beam under bending. Both of these methods express shear and moment in terms of rotation and deflection at two ends of each beam or a segment of • a beam, and the problems for analysis are formulated by satisfying equilibrium conditions at junctions of adjacent segments. While the slope.deflection method solves directly the equations established from equilibrium conditions, the moment distribution method follows an iterative procedure. Detailed discussions on these methods can be found in textbooks on structural analysis 13. The original ideas behind these methods may be found in References 4-7. For the moment distribution method, Gere 8 gives the most extensive treatment. Wang and Salman 3 give brief and concise presentations on the historical developments of these methods. The slope-deflection method in matrix format for analysing statically indeterminate structures has become more emphasised as computers are increasingly used. While the slope-deflection method is well established and widely used, derivation of the basic equations based on the superposition principle is cumbersome. Furthermore, the indeterminate fixed end moments involved must be determined using a separate method. These shortcomings may be removed by considering directly the general solution for the bending of beams from the basic structural mechanics points of view. Also, in some cases, the use of quantities other than slope and deflection as primary unknowns at each end may become more desirable. These are the reasons which motivated this study. In what follows, explicit expressions for a new set of slope and deflection equations will be presented first. As an alternative procedure, the bending moment and deflection at each end of a beam or a segment of equations established by requiring continuity in rotation at each joint in conjunction with force equilibrium conditions is solved for the unknown end
moments and deflections. The method may be called the moment-deflection method. Since the moment is directly proportional to the curvature of the deformed beam, the method may also be referred to as the curvature-deflection method. Although the slopedeflection method is well established and widely used, the present moment-deflection method is comparable to the slope-deflection method in concept and at times may be easier to apply. In the conventional slopedeflection method, determination of fixed end moments requires an analysis of an indeterminate problem itself. Consequently, only simple loading conditions are routinely used. The present method, however, provides direct solution without relying on other supplementary procedures. Furthermore, the moment.deflection method may be applicable to structures having semi. rigid joints. w
I
.~F----T
M~.~V+dV
P
w~ X
0
- ~ d xl~-
b
Figure I Geometry and =ign convention
Basic equations The deflection, w, of a beam under a loading function p(x) is governed by the following well known equation -Elw"
=p
(1)
in which a prime denotes differentiation with respect to x, which is the coordinate along the longitudinal axis. The shear V and bending moment M are V --- E l w "
(2)
M = Elw"
(3)
respectively. The geometry and sign convention are shown in Figure 1. The general solution of equation (1) is Elw = A w 3 + Bx 2 + Cx + D + P(x)
(4)
in which "School of Civil Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
140
0950 - 0618/90/030140 - 07 © 1990 Butterworth-Heinernann Ltd
P(x) = I~ p(x)dxdxdxdx
(5)
3 CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
The loading p, which may involve singularity functions, is considered to be explicitly integrable. Otherwise, certain series representations may be used. The integration constants A, B C and D may be expressed in terms of either end slopes and deflections, or end • moments and deflections.
New slope-deflection equations By choosing slope, 0 = w ', and deflection, to, at ends a and b of a beam segment of length L as the independent quantities, i.e., w(0)=w,, w(L)=%,
w'(O)=O,, w ' ( L ) = O b
(6)
than the conventional slope-deflection equations. As an example for illustrative purposes, the function P(x) for the following class of loading p = -q(x/L)"
(15)
P(x) = - [n!/(n + 4)! ]qx"+4/L"
(16)
is
As a result P(O) = P'(O) = P " (0) = 0
(17)
P(L) = - [ n!/(n + 4)!]qL'
(18)
P ' ( L ) = - [n!/(n + 3)[]qL 3
(19)
P " ( L ) = - [n!l(n + 2)!]qL 2
(20)
the integration constants in equation (4) are found to be A = El(Oh -- O,)/L 2 - 2EI(wb - w , ) / L 3 -
+P'(O) ] IL 2 - 2 [P(O) - p ( L ) ] / L a
[P'(L) (7)
B = - E I ( 2 0 , + Ob)/L -- 3 E I ( % - w , ) / L 2 + [2P'(O)
+ P'(L)]/L + 3 [ / ' ( O ) - P(L)]/L 2
(8)
- e'(0)
(9)
D = E l w , - P(O)
(10)
C = E/O,
P'(L)
= - [1/(n + l ) ] q L
(21)
By substituting equations (17) to (21) into equations (11) and (12) for the fixed end conditions with w. = wb = 0, = #b = O, the fixed end moments are found to be ME° = 2 P ' ( L ) / L - 6 P ( L ) / L 2
= - [2(n + 1)!/(n + 4)!]qL 2 The end m o m e n t s M, and bob at x = 0 andL respectively from equation (3) b e c o m e the new slopedeflection equations as follows M , = - 4 E I O , IL - 2EIOblL + 6El(wb -- wo)lL 2 + MF,
(22)
MR, = P " ( L ) - 4 P ' ( L ) / L + 6 P ( L ) / L 2
= - [(n + 2)!/(n + 4)!]qL 2 Table I
(23)
Fixed end moments for v a r i o u s values o f n
(11) Mb ffi 4EIOb/L + 2EIO,/L - 6 E 1 ( % - w , ) / L 2 + M R ,
0
1
2
10
1/12 1 / 12
1/30 1/20
1/60 1/30
1/1092 11182
(12) - MF° I(clL 2) - MFb I(oL 2)
in which MF° = P " ( O )
+ 2 [2P'(0) + P'(L)]/L
+ 6 [P(0) - P(L)]/L 2
(13)
MR, = P " ( L ) - 2 [2P'(L) + P ' ( O ) ] / L
- 6 [P(O) - P(L)]/L2
(14)
The luted end moments corresponding to various values of n are given in Table 1, where the results for n = 0 and 1, corresponding to uniform and linearly varying loadings, are seen to agree with the well known exact solutions. For sinusoidal loading p = q s i n ( m x x / L ) , t h e function P(x) is give by P(x) = q L ' / ( m x ) 4 s i n ( m w x / L )
Except for the sign conventions for 0,, 0b, W, and wb which may differ from author to author, equations (11) from which and (12) are in the usual form of slope-deflection P(O) = P ( L ) = P"(O) = P"(L) = 0 equations, in the conventional slope-deflection method, a separate analysis method is needed to solve the P'(O) = p m (0) = qL3/(m'lr) 3 statically indeterminate problem for the Fu
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
(24)
(25) (26) (27)
(28) (29)
141
If the loading function is represented by a Fourier sine series, then the summation for m ranging from 1 to Qo should be made. For a beam under a concentrated load F at x = and a moment M at x = ~ the loading function expressed in terms of a singularity function x - ot n is
E/O b = El(Wb -- wa)/L +
(2Mb + Ma)L/6 + P'(L)
- [2P"(L) + P"(O)]U6 -
[P(L)
-
P(O)]/L (44)
V= = (M~ - Mb)/L + [P"(L) - P"(O)]/L
-
Pro(0) (45)
p = F(x-
~) -j + M ( x -
(30)
vl>-2
Vb = ( M = -
Mb)/L + [P"(L) - P"(O)]/L - P#'(L) (46)
l:or which P(x) = F ( x - ~)3/6 + M ( x - ~1) 212
(31)
P(O) =P'(O) =e"(0) = 0
(32)
P(L) =F(L - ~)3/6 + M ( L - ~)2
(33)
P'(L) = F(L
~)2 + M(L - 77)
(34)
~) + M
(35)
P"(L)
-
=F(L-
At each joint, the end m o m e n t s of adjacents m e m b e r s are related through the m o m e n t equilbrium condition. By requiring continuity in rotation at each joint together with additional force equilibrium equations, one then establishes a system of simultaneous algebraic equations for the unknown m o m e n t s and deflections. We refer to this procedure as the moment-deflection method. The m e t h o d may best be illustrated by examples.
Examples
The fixed end m o m e n t s b e c o m e MF, = k(1 - k)2FL + 2(1 - s)M - 3(1 - s)2M
(36)
Mro = E(1 - k)FL + M - 4(1 - s)M + 3(I - s)2M (37) in which k = ~/L and s = ~I/L. Based on equations (11) and (1 2) as the new slopedeflection equations, the procedure for structural analysis follows that of the conventional slope.deflection method.
In order to illustrate the moment.deflection method and to make comparisons with the slope-deflection method, the following examples given in the textbook by Wang and Salman 3 for illustrating the application of the slope-deflection method to statically indeterminate beams and frames are considered. It should be noted that the sign convention used in the present method of analysis is different to that used in Reference 3. E x a m p l e 9.3.1 o f Reference 3: A n a l y s e the t w o s p a n continuous b e a m s h o w n in Figure 2. q= 2 k i p s / f t
Moment-deflection equaUons and procedure By choosing M and w at ends a and b of a beam segment of length L as the primary quantities, i.e.,
h~rrB
Lt = 2 0 f t -
[=
constant EZ L z = 30 ft
:i--
:'~
w(O) = w,, w(L) = % , Elw"(O) = M , ,
(38)
EIw"(L) = Mb
the integration constants A, B C and D in equation (4) are found as follows A = {(Mb -- M a ) / L - [P"(L) - P"(O)]/LI/6 B = [M=-
-
Figure 2
(39)
(4O)
P"(0)]/2
C = E1(Wb - w~)/L
25
t
Example 9.3.1
F = 2 4 kips
,
,
Iq, = 0 . 9 k i p s / ~ , %
A 21 ~1~_ kL_~,.~,=~_i0 f t ]= Ll:16ft
(Mb + 2M~)L/6
4
qz = Z k i p s / f t
÷
= ==
- [P(L) - P(O)]/L + [P"(L) + 2P"(O)]L/6
,
(41) D = EIw= - P(O)
(42)
The rotation and transverse shear force at a and B expressed in terms of w,, wb, Mo,/v/b and the loading are considered as the moment-deflection equations ElOa = El(wb -- wa)/L
-
(2Ma
-
P(O)
]/L
(43)
142
2 3 Figure 3
4 5
Example 9.3.2
.Solution. Since there is no m o m e n t at A and C, and no
deflection at A, B and C, only one condition, 02 = 03, needs to be satisfied. Using the moment.deflection equations, one obtains
+ Mb)L/6 + P'(O)
+ [ P" (L) + 2P" (0) ] L/6 - [ P(L)
5] Lz= 2 4 f t
EI02 = M2LI/3 + qL~/24
(47a)
El03 = - M ~ . 2 / 3 - qI.~124
(47b)
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
Equating equations (15) and (16) and knowing that h/2 = M~ because of moment equilibrium at the joint B, one obtains for q = 2 kips/ft, L~ = 20 ft and L 2 = 30 ft,
= -(q/S)(L~ = - 175 ft-kips
M2 =
M~
+
I.~)I(L,
+ L2)
(48)
which agrees exactly with the solution given in Reference 3. For this example, only one equation is used to solve the problem while three simultaneous equations for 0^, 0B and 0c are involved when the slope. deflection m e t h o d is used. Clearly, if ends A and C are fixed, then the present method would involve three equations and only one equation would be involved for the slope-deflection method.
Example 9.3.2. of Reference 3: Analyse the continuous beam shown in Figure 3. Solution: Using the numbering in Figure 3 and apply-
ing the moment.deflection equations
slope.deflection method involve two simultaneous equations.
Example 9.4. I of Reference 3: Compute the midspan deflection of a simply supported uniformly loaded beam as shown in Figure 4 utilising a joint at the midspan. Solution. Using the numbering in Figure 4 and apply-
ing the moment-deflection equations
(49)
El, 02 = (2M2 + M,)L,/6 - ql L3/24 -FL2k(1 - k2)/6
(56)
EI03 = -2EIw2/L - M3L/6 - qL3/192
(57)
V2 = -2M2/L + qLI4
(58)
V3 = 2M3/L - qL/4
(59)
By knowing that /4~ = /43 from the m o m e n t equilibrium and w2 = w3 = we, the continuity in rota. tion 02 = 03 and the force equilibrium V2 - V3 = 0 at joint B lead to the following two equations
EllOi = -(2MI + M2)Lx/6 + qlL~/24 + Ffk(1 - k)(2 - k)/6
E102 = 2EIw2/L + M2L/6 + qL3/192
4EIwBIL + M2L/3 = -qL3196
(60)
M2 = qL2/8
(61)
from which one obtains the well known result
(50) (62)
EIwB = - (5/384)qL 4 E1203 = -(2M3 + M4)/.,zl6 + q2/-~/24
(51)
From the moment equilibrium at joints A, B and C and by knowing that/45 = -q2L~z/2, one has M2 =M3 and M4 -- M5 = - 3 6 ft-kips
(52)
The present method involves two equations for deter. mining we and M2 while the slope-deflection method would involve four equations for 0^, 0s, 0c and we. Clearly, if the beam is fixed at both ends, then the present method would involve for equations while the slopedeflection method would involve only two.
q kips/ft
r~,~ L"=Lz-I ~ F =40 kips B
A l"
B L
I
C ,1 I
I
23 B
Figure 4
q
2
Lz
for q2 = 2 kips/ft and L 3 = 6 ft. With q~ = 0.9 kips/ft, F = 24 kips, k = 3•8, Ll = 16 ft, L2 = 24 ft, II = 21, and/2 = Is = 51, the two required conditions 0~ = 0 and 02 = 03 lead to the following two equations
M~ + 3.2M2 = - 3 3 2 . 5 5 ft-kips
L~
4
Example 9.4.1
2M~ + M2 = - 2 0 3 . 8 5 ft-kips
3
C
51
Figure 5
Example 11.7.3
14
I
Example II. 7.3 of Reference 3: Analyse the rigid frame having two hinged supports as shown in Figure 5.
(53)
Solution. The axial coordinate from A to B, B to C, and C to D is considered to be in the positive direction. Us(54) ing the numbering in Figure 5, the moment deflection equations give
Solving equations (22) and (23), one obtains
Ell 02 = Ell w2/Ll + M2LI/3 + qL~/24
(63)
I/2 = -M2/Lm + qL,/2
(64)
M l = - 5 9 . 2 1 7 ft-kips and M 2 = - 8 5 . 4 1 7 ft-kips
(55) which agree exactly with the results given in Reference 3. For this example, both the present method and the
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
E1203 = -(2M2 + M , ) ~ / 6 - FI~k(1 - k)(2 - k)/6
(65)
143
El204
-~"
(2M4 + M3)L2/6 + F ~ k ( 1 - k2)/6
EI305 = - E l a wsIL3
-
(66) (67)
MsL313
Vs = M5/L3
(68)
Clearly, M2 = M3, M4 = M5 and w2 = - w s . The conditions to be satisfied are V 2 + V 5 = O, r M 3 + E1302 = El303
and 04 = 05 (69)
in which the parameter r is introduced to account for the semi.rigidity of joint B. If the joint is rigid, r = 0. Using equations ( 6 3 ) - ( 6 8 ) with q = 0.5 kipsflt, L] =
L3 = 16 ft, L2 = 20 It, k = 0.4, 11 = 31, 12 ffi 51,/3 = 21 and r = O, the conditions given in equation (69) result in the following equations
handbooks, the use of the conventional slope-deflection method is normally limited to such simple loading conditions. While the new slope-defiection and momentdeflection methods are more direct and flexible than the conventional slope-deflection method, the loading function must be explicitly integrable. For cases when loading functions are not explicitly integrable, a method using Fourier series in conjunction with the Stokes' transformation is presented here. Dicussions on the Stokes' transformation used in the Fourier series analysis can be found in Reference 9. Representing any loading function Pm sinot,nX for 0 < x < L
p(x) = ~
(74)
m~l
and the displacement function oa
M3 - M4 = 64
(70)
- ( 9 / 1 6 ) E 1 2 w s + (9r + 140)M3 + 30M4
=
for
O
IV,
at
x = 0
Wb
at
x = L
-10496 (71)
w(x) =
(72)
(3/16)E12w5 + IOM3 + 60M4 = - 2 6 8 8
Solving equations ( 7 0 ) - ( 7 2 ) for r = 0 M3 = - 1 3 . 4 7 4 kips-ft, M4 = - 7 7 . 4 7 4 kips-ft
and
Elws = 2234.796 kips-ft 3
(73)
which agree with the results given in Reference 3. Since the frame is supported by hinges at A and 13, the present method involves fewer equations than the slopedeflection method. While the slope.deflection method cannot account for the semi-rigidity of the joints, it is a simple matter to include this effect in the present method. To show the effect of r at joint B, some numerical results are given in Table 2. It may be noted that r ranging from 10 to 0o corresponds to the joint varying from rigid to hinged connections respectively. It is seen in Table 2 that the moment at joint B reduced from the rigid joint case by about 50% when r = 40 and by over 90% when r = 400. Table 2
W~sinctmx
Effect of semi-rigidity of joint B (M 4 = M 5 = M 2 -
(75)
in which ¢=m= m x / L and Pm and Wm are Fourier coefficients for the loading and displacement functions respectively, the derivatives of the displacement function given in equation (75) following the Stokes' transformation are
w'(x) =
(Wb -
W,)IL+ ~ W'=cosot,x m=l
for 0 _< x < L
(76)
where W'm = t~, W., - (2/L) [ W. - ( - 1)mWb]
t~,~W" s m t ~ x
for 0 < x < L
64)
w"(x)= r
M 2 = M 3
r
M2 = M 3
0 10 20 30 40 50 60 70 80 90 1O0
- 13.474 - 10.894 - 9.143 - 7.877 -6.919 - 6.169 - 5.565 - 5.069 -4.655 - 4. 303 - 4.000
150 200 300 400 500 600 700 800 1,0OO 10,000 1O 0 , 0 0 0
- 2.960 - 2.349 - 1.662 - 1.286 - 1.049 - 0.886 -0.766 - 0.675 -0.546 - 0.057 - 0.0057
at
x -- 0
at
x =L
(77)
and ao
w " (x) = (Kb -- K=)/L + ~ ,
W~' cost~=x
m=|
for O - - x _ < L
(78)
where
Fourier a n a l y s i s with Stokes' transformation Since the fixed end moments corresponding to simple loading conditions are commonly given in textbooks or
144
W," = (2/L)ot2 [ W, - ( - ( 2 / L ) [x= - ( -
1)mWb] 1)%b] - cx~Wm
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
L. The beam is under uniform load q distributed trom x = 0 to k L for 0 k 1. The exact solution for the fixed end moment at x = 0 is
and
w'~(x) = - ~
t~,~WZ sinctmx
m=l
(79) M(O)
'
ln the above, Wo and W~ are deflections at ends a and b, and Ko and Kb are the end curvatures. By substituting equations (74) and (79) into the governing equation (1) for the beam deflection, and collecting like terms in the resulting equation, the unknown Fourier coefficients for displacement Wm are expressed in terms of the end curvatures K, and K~, deflections W° and W~ and the loading coefficients p,,, as follows
=
(87)
~qL=/8
in which /~ = I - ( 1 - k)212 - (I
-
k)2]
The corresponding solution for the fixed end moment /v/s at x = 0 based on the Stokes' transformation using the curvature.deflection method is found to be
Ms(O) =aqL2/8 W . = (p./n,~)/EI + (2/L)I~. [ W= - ( -
1)'Wb ]
-(2/L)/~m [~, - ( - 1)%b]
(80)
(89)
in which o=
ce = (48hr 4) ~ It is interesting to note that the general solution is obtained in terms of end curvatures and deflections, which indicates that the moment.deflection method presented earlier appears to be more natural than the conventional slope-deflection method. The slopes and shear forces at the end sections expressed in terms of Wo, Wb, Ko and K~ become 0. = (Wb -- I4,'=)//., -- (L/3)K= -- (L/6)Kb OD
(81)
(88)
[ 1 - cos(/o'nT) ]/m
(90)
m=|
By taking 60 terms in the series for
Ob
(Wb -- W,)/L + (L/6)~<. + , ( L / 3 ) K
b
/~ = k 3 [ 1 + (1 - k)3x2/(3k)
o0
+
(82)
(mffil
Letting Wo = Wb = 0, = Ob = 0, we obtain the following fixed end moments for a beam
+0r2/2)(1
M,. = EX,. = ( 2 / L )
[2 + (-l)m](p=/=~)
go
[I + 2 ( - 1 ) ' ] ( p ~ a ~ ) (84)
~ p,./CX,.
~_, (k/m3)sin(kmx)/(1 - k2m 2)
(94)
m ffi l
m=l
K b ) / L -F
(93)
(83) in which c~ = 2hr
MFo = E&b ffi - ( 2 / L )
(92)
The corresponding solution for the fixed end m o m e n t M. at x = 0 using the curvature.deflection equation with the Stokes' transformation is found to be
m==l
E/(Ka -
k)(1 - 2k/3) + (1 - k)]
Ms(0) = o~(3qL2/r 3)
m
Va =
--
(85)
It should be noted that when rn = 1/k, the term in equation (94) must be determined from a limiting process. For such cases when rn = 1/, the term is given by
l,~mI [ sin(km~)/( l - k2m2) ] = ~r/2
(95)
,'O=1
As a result, equation (95) b e c o m e s Vb -- E/(K a -
Kb)/L -F ~
(-- l)"*pm/ol.
(86)
m=l
ot = k 4 + 21~r
(k/m3)sin(km~r)/(1 - k2m 2)
(96)
m=l
The subsequent procedure for analysis will be the same as that of the curvature-deflection method. For illustrative purposes, two examples are presented. The first example considers a uniform beam of length L which is fixed at x = 0 and simply supported at x ffi
CONSTRUCTION & BUILDING MATERIALS Vol. 4 No. 3 SEPTEMBER 1990
km# l By taking 60 terms in the series for ¢xin equations (96) for various values of k ranging from 03 to ], o~agrees with/3 given in equation (94) to the fourth digit.
145
Conclusions A new set of slope-deflection equations is presented that can be used more directly than the conventional slope. deflection equations. An alternative procedure referred to a s the moment- or curvature.deflection method, • which expresses joint rotation and shear force in terms of end moments and deflections, is discussed. In the moment.defiectin method, a system of equations is assembled by requiring continuity in rotation together
with equilibrium conditions. The method, similar in concept t o t h e s l o p e - d e f l e c t i o n m e t h o d , can i n c o r p o r a t e t h e effect of semi-rigidity of the joints. When the loading f u n c t i o n is n o t e x p l i c i t l y i n t r e g r a b l e , t h e m e t h o d b a s e d o n t h e F o u r i e r a n a l y s i s w i t h S t o k e s ' t r a n s f o r m a t i o n is presented, and the procedure follows that of the momentor curvature.deflection method.
References 1 Hsleh, Yuan-Yu. Elementary Theory of Structures, 2nd edn. Prentice Hall Inc., Englewood Cliffs, N J, 1982 2 Maugh, L C. Statically Indeterminate Structures, 2nd edn., Wiley,
blew York, 1964 3 W a n g , C K and 8 a l m a n , C G. Introductory Structural Analysis,
Prentice Halt Inc., Englewood Cliffs, blJ, 1984 4 Crolm, H. 'Continuity as a factor in reinforced concrete design', Prec., ACI 1929, 25, 669.708 5 Cross, H. 'Simplified rigid frame design, Report of Committee
301, Prec., ACI, 1929, 2 6 6 M e n d e r l L H. 'Die Berechnung der Sekundarspannungen, Welch im einfachen Fachwerke in Folge Starter Knotenverbindungen auftreten', Forster's Bauzeitung 1880, 4 5 , p,34 7 Mohr, O. 'Die Berechnung der Faehwerke mit Starren Knotenver. bindungen', Der Civiltngenleur 1892, 38, 577-594; 1893, 39,
67.70 8 Gere, J M. Moment Distribtuion, Van blostrand Rheinhold Com-
pany, Princeton, blJ, 1963 9 (:hung, H. 'Free vibration analysis of circular cylindrical shells,
J. Sound and Vibration 1981, 74 (3), 331-350 Notation
E I M p r V
w x
0
modulus of elasticity area moment of inertia bending moment transverse loading parameter for semi-rigid joint transverse shear force transverse displacement axial coordinate rotation
Acnowledgement This paper was originally presented in Engineering Structures Vol.12, July, 1990 and is reproduced here by kind permission.
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