Application of the Method of Least Work to redundant structures (Airplane Section report)

              File 1> 52.16/64
Vol. V
NON-CIRCULATING
<AVIA
PUBLISHED BY THE CHIEF OF Al ,
McCOOK FIELD REI'ORT, SERIAL No. 2420
D. C.
No. 495
APPLICATION OF THE METHOD OF LEAST WORK
TO REDUNDANT STRUCTURES
( AIRPLANE SECTION REPORT )
\J ',
Prepared by
C. J. Rowe, Engineering Division, Air Service
McCook Field, Dayton, Ohio .
November 6, 1924
WASHINGTON
GOVERNMENT PRINTING OFFICE
1925
Ralph Brown Draugho
. LIBRARY n
MAY 17 2013
Non-Depoitory
Auburn University
INDEX
Introduction and procedure ___ ________ ______ __________ ___________ -- --- ---- --- ------------- - - --
Dimensional properties of members:
Thble 1-Lift truss members _______ _________________________________ _______ ___ ___ _ -- ---- - -- -
Table 2-Upper wing- Drag truss members ___ -------- - ____________ ----- - ------- - ______ ---- ---
Table 3-Lower wing-Drag truss members _____________ --------- ________ ------ ______ - - - - -----
Stress analysis of outer bay _ ___ ___ ___ ___ ________ __ _______ _____ - -- - - -·- _ - - - - - - - - - -- - - - - - - - - - -- --
Work equations and tabular method of writing them ____ ______________________ ___ __________ _____ _
Computation of numerical values to be used in work equations ______ ~ ___ ________ ___ ____________ - - -
Computation of P 0 for inner bay __ ________ ______________ _____ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Computation of PP----------- - - - ~------------ -- ------- -- ---- -- --- - - - ----- - - - ------- -- ---- - -
Computation of PR------ ________ ___ ___ ---- - _____ - -- - - - - - - -- - -- - -- - ---- -- - - - - - -- - - - - -- - -- - - -
Computation of Ps-- ------- --- --- ---- --- ------- ---- -------------- -- ------- --- ------------ --
Computation of Py _________ _____ ___ .. ___ ________________ ______ ~ ____ ___ _____ - - - - - - - - - - - - - - - - -
Computationof Pw- ______ ____ ____ - - ___ - __ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Computation of Pz--- -------- ---------- ---- ---- ---- ------------------------------ ------ ----
Table 4-Summation of values required for the solution of work equations on page 3 ______ _ - - - - - - -
Solution of work equations, suggestions for simplifying _ __ ___ ___ ______ _____ __ ___ ___________ ___ _ - - -
Table 5-Tabular method of solving "lork equations _______ ____ ___ ____________ __ __________ - - - - -
Table 6-Final summation of stresses ___ - ------------- _________ ________________ ______ - -- _ - - - -
Conclusions _________ ____ ____ ____________________ ____ __ ____________ ________ ______ - ___ - - - - - -·- -
Center line drawings of trusses __ __ _ -------------- __ -~ ------ ___ ______ ___ __ --- -~ _______________ _
Fig. 1-Front elevation-Lift truss ____ _______ ___ __ ___ ________ ___ ___________ ___ _______ ______ _ _
Fig. 2-Drag truss-Upper wing ___ ___ ------- - --- - __ - ------ - -- ________ ____ ___ __ __ _____ __ - - -- -
Fig. 3-Drag truss-Lower wing ________ ~ --------- ___ ____ ______ ____ ___ ______ __ _______ _______ -
Fig. 4--Plan view showing external drag members _____ ____ ___ ___ ________ ___ _____ ___ _______ ___ _
Fig. 5-End view, external drag members _________ ____ ______ ____ ___ _____ ___ _______ --.- ____ - - - - -
Fig. 6-Wing truss showing members of statically determinate buy ____ __ _____ ____ _____ - - - - - - - - - -
Fig. 7- Members of the statically indeterminate inner baY-- - - - --- - ---.- --- - ---- -- - - - - -------- -- -
Fig. 8-Inner bay without redundant members _____________ __ ___________ ________ ____ --- _ - - - - - -
(III)
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APPLICATION OF THE METHOD tlf LEKST W_;!l)RK TO
RE.DUNDANT STRUC . -: ~SJ
INTRODUCTION
The purpose of this report is to show an applica­tion
of the Method of Least Work to multiply redun­dant
structures in a manner which will be concise and
will show an efficient method of attack and tabulation
of results. The development of the work equations
and their solution will be carried out with the aid of
a method of tabulation which the writer believes to
be more efficient than those usually described in con­nection
with the Method of Least Work. The purpose
pf this metl!tod is to reduce the labor required to solve
several simultaneous equations and thereby avoid
chances for confusion or the introduction of errors.
The framework of the U. S. A. C- 11 or Lepere
airplane wings is used for an eX-a:mple in this report
and an analysis of the structure in. the low incidence
flight condition is carried through in detail.
It is assumed t .hat the reader is already familiar
with the principles of Least Work. However, where
this is not the case, he is referred to McCook Field
Report No. 2212, Chapter II, Airplane Design, Articles
35 to 37, inclusive.
PROCEDURE
In beginning any analysis, the structural members
of the framework should always be located on fully
dimensioned line drawings. Examples of these draw-ings
are shown in figures 1 to 5, inclusive. With this
information, such tables as 1, 2, and 3, are compiled,
giving the properties of the various structural mem­bers
and the values of their components in the planes.
Other drawings should be made showing the magni­tude,
direction and position of application of all
external forces which will induce stresses in any of the
members of the structure under consideration. These
drawings a·re illustrated in figures 6 to 8, inclusive.
The computation of these external loads is not within
the province of this report, but they can be found in
McCook Field Report, Serial No. 2425.
In the analysis of any redundant structure by the
Method of Least Work, the stresses must be carried
through the members of the redundant truss into some
:fixed point of structure. In the analysis of the Lepere
wings, the fuselage was used as the fixed support.
in this wing framework only the inner bay as shown·
in Figure 7, was redundant. The outer bay, however,
had to be analyzed by the ordinary methods of statics
to obtain some of the loads on the·inner bay.
Where the stress in a member was found by the solu­tion
of a single equation, the member in question was
noted in the margin at·the left af the equation. This
was found to aid in back reference for checking and
tabulating results. This method is shown in the
following stress analysis of the outer bay.
TABLE 1.- Lift truss members
Mem­ber
Material E Area
Q=
L/AEX
10 6
Component lengths Components
Length 1---~--~---1---~------
Beam•• Chord** Side Beam Chord Side
------ - - - - --------1--- - --1---- ------------------- ---
A* L=4. 75, D=l.188 __ _______ ___________ _
B* L=4. 75, D=L 188 _____ ______ _________ c• L=5.125, D = l. 562 ____ ____ __ __ _______ _
D* L=4. 75, D=L 188 __ __ ___ ____ __ _______ _
E 3. 82 X 1.188 _______ ___ _____ _________ _ F* L=4, D=l. 375 _________ ___________ ___ G 2;. in. cables ___ _____________ ______ ___ H 2+, in. cables ______________________ ___ _
I 2+, in. cables.- ------- ----- "------- - -- - J 2Ys in. cables _______ __ ___ ___________ __ i I JI 1~· E~IE~~~-:~/~~~E~iiiiii~~~~~ p· 11'.. in. cables ___________ ______________ _
Z 2-h in. cables __________ ______ ____ ____ _ R 2-h in. cables ____ ________________ _____ _
S 1-i\ in. cables ___ ____________ ___ __ _____ uT*• LL==64., 2D5,= Dl.= 3l7.5 3.7.5. -_-_-__-_--_-_-_-__-_-_-_-_--__--__-_- -_- V 1. 75 X 0.125 steel__ ___________________ _
X 1-i\ in. cable _________________________ _
Y 1-h in. cable ______ ______ _____________ _
• Streamline struts, spruce.
All dimensions given in inches.
1, 600, 000
1, 600, 000
1, 600, 000
1, 600, 000
I, 600, 000
1, 600, 000
25, 550, 000
24, 870, 000
24, 870, 000
27, 728, 200
25, 550, 000
26, 195, 000
26. 195, 000 I 24, 870, 000
25, 550, 000
25, 550, 000
26, 195, 000
26, 195, 000
26, 195, 000
1, 600, 000
1, 600, 000
28, 000, 000
26, 195, 000
26, 195, 000
4.1194
4. 1194
5. 8437
4. 770
4. 520
4. 015
0. 0576
0. 0344
0. 0344
0. 0178
0. 0576
0. 0412
0. 0412
0. 0344
0. 0288
0. 0288
0. 0412
0, 0412
0. 0206
4. 015
6, 270
0. 2187
0. 0206
0. 0206
•• See Part I, Serial No. 2332 and Chapter V, Airplane Design.
(1)
9.40
9. 48
6. 63
8.19
8. 64
9. 65
78. 5
135. 0
135. 0
237. 5
67. 5
92. 0
92. 0
116. 0
134. 5
148.1
70. 5
78.0
95. 5
5. 37
3. 69
8. 32
61. 15
92. 66
62. 0
62. 5
62. 0
62.5
62.5
62. 0
115. 5
115. 5
115. 5
115. 5
99. 25
99. 25
99. 25
99. 25
99. 0
109. 0
76. 0
85.0
51. 5
34. 5
37. 0
51.0
33. 0
50. 0
56.4
56. 9
56. 4
56. 9
56. 9
56. 4
56. 5
56. 5
56. 5
56. 5
56. 5
56. 5
56. 5
56. 5
58. 0
58. 0
31. 5
31. 5
41. 0
30. 0
30. 0
0
26. 5
26. 5
25. 0
25. 25
25. 0
25. 25
25. 25
25. 0
26. 0
26. 0
26.0
26. 0
26. 5
26. 5
26. 5
26. 5
15. 312
49. 562
3. 27
37. 2
30. 0
15. 5
16. 6
0
19. 7
42.5
0
0
0
0
0
0
97.0
97. 0
97. 0
97. 0
77. 0
77. 0
77. 0
77. 0
80. 0
80. 0
69. 6
69. 6
7.4
7. 3
7. 7
51. 0
0
0
.910
. 910
. 910
. 910
. 910
. 910
. 490
.490
. 490
.490
. 570
. 570
. 570
.570
. 585
. 532
. 415
. 324
. 796
. 885
. 810
0
.803
. 530
. 404
.404
. 404
. 404
. 404
.404
. 225
. 225
. 225
. 225
. 268
.268
.268
. 268
. 155
. 455
. 043
. 402
. 583
.448
. 450
0
. 596
.848
0
0
0
0
0
0
.840
.840
.840
.840
. 775
. 775
. 775
. 775
. 805
. 732
. 918
. 745
.143
. 210
. 208
1
0
0
2
TABLE 2.- Upper wing-Drag truss members
•r Q= Component lengths Components
Mem- ber Material E Area L/AEX Length 10• Beam Chord Side Bean Chord Side
ab Spruce ____ ____ ____ ___ __ _._ ,- _____ ____ __ 1, 600, 000 1.554 12. 85 32. 0 0 32.0 0 0 1 0
cd Spruce . _____ _____ -- - - -- - -- ------ -- - -- - 1, 600, 000 1.554 12. 85 32.0 0 32. 0 0 0 I 0
tf Spruce __ ____ _______ ________ _________ __ 1, 600, 000 1.930 10. 35 32.0 0 32.0 0 0 I 0
gh Spruce ___ ___________________________ __ I, 600,000 1.930 JO. 35 32. 0 0 32. 0 0 0 1 0
ij Spruce __ _____ ______ ______ _____ _______ _ !, 600, 000 2. 872 6.95 32.0 0 32. 0 0 0 I 0
be lo-J2 tie-rod ___ ____ ___ _______ ------ ____ 28, 000, 000 o. 0141 154. 50 61. 0 0 34. 2 50. 5 0 0. 561 0. 828
de 10-32 tie-rod_ __ ___ _____ __ __ __ - -- --- ____ 28,000, ()()() 0. 0141 150. 50 .'>9. 5 0 34. 2 48.0 0 o. 575 0.806
fg 10-32 tie-rod _____ __ __ _____ ____ _ - - - - ____ 28,000, ()()() 0. 0141 132. 80 52. 5 0 34 .. 2 39. 5 0 o. 650 o. 752
hi 10-32 tie-rod __ __ -- - ----- ______ : ____ ---- 28, 000,000 0. 0141 130. 50 51. 5 0 34. 2 38. 5 0 0. 665 0. 748
I Spru~-e- _____ ____ ____ ______ ____ ___ ___ __ 1,600,000 4. io 7. 70 50. 5 0 0 50.,., 0 0 l
m ~g~~i::::::::::::::=:::::::::::::::: . 1, 600, 000 4.10 7. 32 48.!f 0 0 48. 0 0 0 I
n I. 600,000 4.10 6.03 39. 5 0 0 39. 5 0 0 1
0 I, 600,000 · 4. 10 5. 86 38. 5 0 0 38 . .5 0 0 I
p Spruce ___ ___ ____ ____ ___ ___ -- -- - - __ ____ l, 600,000 4.10 5. 30 34. 75 0 0 34. 75 0 0 I
q Spruce ___ ____ ____ __ _______ __ --------~- l, 600; ()()() 3. 50 9.02 50.5 0 0 50. 5 0 0 1
Spruce_ --- __________ ------ -- -- ---- - - - - 1, 600, 000 3.50 8. 55 48.0 0 0 48.0 0 0 1
Spruce _________ --- - _ - _ - - -_ - -- - - - - - - - - - I, 600, 000 3. 50 7. 05 39. 5 0 0 39. 5 0 0 1
Spruce ____________ ______ ---- - - -- __ ____ 1, 600, 000 3.50 6. 87 38. 5 0 0 38. 5 0 0 1
" Spruce ____ __________ __ ________ __ ___ . __ l , 600, ()()() 3.50 6.20 34. 75 0 0 34. 75 0 0 1
TABLE 3.-Lower wing-Drag truss members
I - I Component lengths I Components
Mem- Q=
ber Material E Area L/AEX Length
10• Beam Chord Side Beam Chord Side
a'b' Spruce_ --- -- __ -- - -- --- - - - - - - ----- - - - -- 1, 600, 000 1. 937 10. 35 32. 0 0 32.0 0 0 1 0
c'd' Spruce ____________________ ___ ________ 1,600,000 l. 554 12.85 32. 0 0 32.0 0 0 1 0
''f Spruce. _______________ -- ___ - - - -- - - -- -- I, 600, 000 !. 930 10. 35 32.0 0 32.0 0 0 l 0
g' '
.Spruce ___________ _________ - -_ -- __ _____ l,600,000 1.930 10. 35 32. 0 0 32.0 0 0 1 0
i'j' Spruce _____________ __ ____ __ --- _______ 1,600,000 2. 872 6. 95 32. 0 0 32. 0 0 0 1 0
b'c' 10-32 tie-rod _____________ -- - ------- ---- 28,000, 000 0. 0141 154. 5 61.0 0 34. 2 50.5 0 0. 561 0.828
d'e' 10-32 tie-rod ________________ ______ . •• _ 28,000, 000 0. 0141 150. 5 59. 5 0 34. 2 48.0 0 0. 575 0.806
f'g' 10-32 tie-rod ________ _______________ --·- 28, 000, 000 0.0141 132. 8 ·52.5 0 34. 2 39. 5 0 0.650 0. 752
h'i' 10-32 tie-rod _________________ --· __ ____ 28,000,000 0. 0141 130. 5 51. 5 0 34. 2 38. 5 Q 0. 665 0. 748
Z' Spruce·------------·-·----- · - ---- ---·- 1,600, 000 4.10 7. 70 50.5 0 0 50. 5 0 0 1
m' Spruce. ____ --· _ ------ _____________ ---- 1,600,000 4.10 7. 32 48. 0 0 0 48.0 0 0 1
n' Spruce. _____________ ___ -- ___________ __ 1, 600, 000 4.10 6.03 39. 5 0 0 39. 5 0 0 1
o' Spruce. ___ ______ ---- ____ -- -- ________ __ 1,600,000 4. 10 5.86 38. 5 0 0 38.5 0 0 1
p' Spruce. _____________ ____ -- ______ ·-- ___ I, 600, 000 4. 10 5.30 34. 75 0 0 34. 75 0 0 1
q' Spruce ___ ___ ___ __________ __ ___________ 1,600,000 3.50 9. 02 50.5 0 0 50.5 0 0 1
r' Spruce. ___ - - -- -- - -- ------ -- ---- --- - - -- 1,600,000 3.50 8. 55 48. 0 0 0 48.0 0 0 1
s' Spruce ____ ______ --- · - __ -- - - - - --- -- - - - - 1,600, 000 3. 50 7.05 39. 5 0 0 39.5 0 0 I
t' Spruce ______ - - ---·--- ---- -· -- ----- -- -- 1,600,000 3. 50 6. 87 38 .. 5 0 0 38.5 0 0 1
u' Spruce ___ - -------- -- -- -- -- - - - · --- - - - - 1, 600, 000 3. 50 6. 20 34. 75 0 0 34. 75 0 0 I
STRESS ANALYSIS OF OUTER BAY (SEE FIG. 6) ' The completion of this preliminary work now leads
to the analysis of the redund~nt inner bay. The first
step is the selection of redundant members. This is a
matter of judgment on the part of the engineer and
no general rule can be given. However, when it
seems probable that certain members will go out of
action, such as wires going into compression, these
members should be chosen as redundant where possible.
A 0.910A+468= 0 A= -5141bs.
B 0,910 B+893= 0 B=-981 lbs.
a'b' a'b'+92 = -514X0.404 a'b' = -300 lbs.
b'c' 0:561 b'c' - 300-981 X 0.404= 0 b'c'=+l,241lbs.
l' l'= O.
q' q1+ I ,241X 0.828 = 0 q' = -1,028lbs.
c'd' c;d' +58+ 1,241 X 0.56i=O c'd' = -754 lbs.
m' m' = 1,241 X 0.828= + 1,028 lbs.
d'e' 0.575 d'e'-754= 0 d'e'= + 1,311 lbs.
r' r'+l,311 X 0.806= - 1,028 r'= -2,085 lbs.
H 0.490 W- 981 X 0.910 == 1,204 H= +4,280 lbs.
G 0.490 G- 514X0.910= 650 G= +2,282 lbs.
ab ab+ 120- 514X 0.404+2,282X 0.225=0
- 425 lbs.
The landing wires of the lift truss were obviously
out of action in this condition of loading and the six
external drag wires were chosen as the redundant ·mem­bers.
The inner bay, without the redundant members,
is shown in Figure 8. The six redundant members
are P, R, S, W, Z, and Y.
ab= The work equation given here is merely an expan-be
0.561 bc ~ 425 = - 981 X 0.404+4,280X0.225
+ l i768 lbs.
l= -2,282X 0.840= -1,917 lbs.
cd cd+76+1,768X0.561 = 0 cd= -1,068 lbs.
m=- l,917 + 1,768X 0.828= -453 lbs.
sion of the general equation with which it is presumed
be= the engineer is already familiar.
W= (Po+PPKP+PRXR+PsXs+PwXw+PzXz+
Q
PvXv) ~
m
de
q
0.575 de-1,068 = 0 de= + l,857 lbs.
q+l,768X 0.828+4,280X 0.840 = 0 q= -5,059
where P0 =the stress due to the applied loads en the
structure made statically determinate by
temporarily removing the redundant mem­bers.
lbs.
r r+l,857 X 0.806= -5,059 r= -6,556 lbs.
,.-
3
Pp, Px, Ps, etc.=the stresses due to a unit tensile load
in each of the redundant members,
P, R, S, etc.
ing the equation. This will avoid
the necessity of working with large
decimals.
Xp, Xx, Xs, etc. = the actual stresses in the redundant
members, P, R, S, etc. These values
are to be found by the solution of
the work equation, Q= L/AE. Since
Q is common to all members of the
equation it can be multiplied by some
quantity such as 10 6 without affect-
The above· equation is reduced to the first degree
and expanded Into the following six equations by tak­ing
the partial derivative of W with respect to ea'cn of
the unknown quantities representing the stresses in the
redundant members and equating these partial deriva­tives
to zero.
The following are .the six equations so obtained as generally written for simultaneous solution:
1. 2:PoPpQ+Xp 2:Pp2Q+XR 2:PPPRQ+ Xs 2:PpPsQ+Xw 2:PpPwQ+Xz 2:PpPzQ + Xv 2:PpPyQ= O
2. 2:PoPRQ+Xp 2:PpPRQ+XR 2:PR2Q+ X s 2:PsPRQ+Xw 2:PwPRQ+Xz 2:PzPRQ + Xv "'i:.PvPRQ = O
3. 2:PoPsQ+XP 2:PpPsQ+XR 2:PRPsQ+Xs 2:Ps2Q+ Xw T.PwPsQ + Xz T.PzPsQ+ X v T.PvPsQ= O
4. 'l:.PoPwQ+Xp 2:PpPwQ + XR 2:PRPwQ+ Xs 2:PsPwQ+Xw 2:Pw2Q+ Xz 2:PzPwQ + Xv T.PvPwQ = O
.5. 'l:.PoPzQ+XP 2;PpPzQ+XR 2:PRPzQ+Xs 2:PsPzQ+Xw T.PwPzQ+Xz 2:Pz2Q+Xv T.PvPzQ = O
6. 2:PoPvQ+Xp T.PpPvQ+XR 2:PRPvQ+X~ 2:PsPvQ+ Xw 2:PwPvQ+Xz 2:PzPvQ+Xv T.Pv2Q= O
The method shown below for writing the above equations in tabular form is recommended as a timesaver
in solving the equations, as the variables do not have to be repeated with each step in the solution and it also
makes the solution of the equations easier to follow. There is a column in the table for the coefficients of each
variable and one for the constants.
No. of equation Constants Xp XR Xs Xw Xz Xv
l _ ------- - - ------------------- - -- - - - -- l:PoPpQ + :i:Pr'Q +:i:PrPRQ +:i:PrPsQ + :i:PrPwQ +:i:PrPzQ +:i:PrPvQ =0
2_ - - - - - - - -- -- --- - - - --- - - - - - -- - - - - - 3 ________ ____ _____ ____ __ ___________- -__-_- l:PoPRQ +:!:PPPRQ +:i:PR' Q +:i:PsPRQ +:!:PwPRQ +:!:PzPRQ + :i:PvPRQ =0 4 ___ ______ ___________ ______ ____ ____ ___ l:PoPsQ +:i:Pr PsQ +:i:PRPsQ +:i:Ps'Q + :i:PwPsQ +:i:PzPsQ +:i:PvPsQ =0 5 _____________ ___________ __ __ __ ____ ___ l:PoPwQ +:i:PrPwQ +:i:PRPwQ +:i:PsPwQ + :i:Pw' Q +:i:PzPwQ +:i:PvPwQ =0 6 _______ __ ___ __ ____ ___ _____ ______ _____ l:PoPzQ +:i:PrPzQ +:i:PRPzQ +i:PsPzQ + :i:PwPzQ +:!:Pz'Q +:i:PvPzQ =0 l:PoPvQ +:i:PrPvQ +:!:PRPvQ +:!:PsPvQ +:i:PwPyQ +~PzPyQ +:i:Pv'Q =0
The foregoing method of tabulating equations is a .
great help in keeping them in order and also in solving
them by the method described below. The values
under the heading "Constants " can be placed on the
right side of the equal sign with their signs reversed.
The value of this will be seen in the solution of the
n' n' - 2080 X 0.752 = 1028+1917+ 1057 n'=+5566
equations.
lbs.
s' s' = - 2085+3595 = .+ 1510 lbs.
g'h' g'h' - 2080X 0.650= 0 g'h'=+1352 lbs.
t' t' = l510 - 2080 X 0.752= - 54 lbs.
h'i' 0.665 h'i' + 46+1352=0 h'i'=-2102 lbs.
L 0.570 L - 3271 X 0.910 = 1216 £=+7356 lbs.
S S+7356X 0.775 + 6556 = 0 S =-12257 lbs.
COMPUTATIONS 01'' NUMERICAL VALUES TO eJ eJ = 7356X 0.268-3271 X 0.404 = +650 lbs.
BE USED IN THE WORK EQUATIONS K 0.570 K - 0.910X 1736= 638 K = +3891 lbs.
The work of analyzing the structure for the values
required in the above equations will now follow in
detail. A sketch for each condition showing only the
members affected by the application of ·a unit load in
the redundant member under consideration should
first be made, as it will help in making the analysis ·and
in keeping it free from errors. These sketches are most
convenient if placed at the head of the computation of
the stresses due to a unit load in the redundant member
in question.
COMPUTATION OF P0 FOR INNER BAY (SEE
FIG. 8)
D 0.910 D+2097+880= 0 D=-3271 lbs.
e'J' e'J'-3271 X 0.404+963=0 e'J' = +358 lbs.
C 0.910 C+1118+462 = 0 C= -1736 lbs.
J'g' ,~ 0.650 J'g' + s2+754+358= -1736 X 0.404+513
J' g' = - 2080 lbs.
fg 0.650 Jg + 650 - 1736 x 0.404 + 1068 + 68 + 3891
X0.268 = 0 Jg= - 3274 lbs.
gh gh - 3274X 0.650 = 0 gh=+2128lbs.
t t= - 1227-3274X 0.752= -14719 lbs.
hi 0.665 hi+60 + 2128=0 hi= - 3290 lbs.
0 0 - 3290X 0.748 = 490 0 = +2951 lbs.
n n+3891 X 0.775 - 3274X 0.752= - 453+1497 n=
+490 lbs.
u u= - 14719- 3290X 0.748= -17180 lbs.
p p = 0 = +2951 lbs.
e 0.910 e=361 e= + 397 lbs.
ij ij+57+0.404X 397 = 0 ij= -217 lbs.
0.910 F+0.803X = 692
-0.404 F+0.596X = 217 + 3290X 0.665 = 2405
F+0.882 X = 760
-F+l.475 X = 5953
X 2.357 X = 6713 X = + 2848 lbs.
F F+0.882 X 2848 = 760 F= -1752 lbs.
O' 0' = 5566+2102 x·o.748=+7138 lbs.
L
$
Compvtal/on o/ Pp
n
0.570 L+0.532 = 0 L= -0.933
S = 0.933X0.775 - 0.732= -0.009
eJ eJ= - 0.933X0.268-0.455= - 0.705
Jg 0.650Jg=0.7.05 Jg=+ l.085
n n= - l.085X 0.752= -0.816
gh gh= -1.085 x 0.650= - 0. 705
hi 0.665 hi=0.705 hi=+l.060
0 0 = - 0.816 - l.060X0.748= -l.609
p p=O= -1.609
t t= - 0.009+1.085X 0.752= +0.807
U U=0.807+1.060X0.748=+l.600
0.803 X+0.910 F= O X = -1.133 F
0.596 X - 0.404 F= - l.060X 0.665
4
U 1.919 U= - L263 U= -0.658
T T=+0.602-0.366= +0.236
V V - 0.658 X 0.208 + 0.236 X 0.210 = 0.745 V =
+0.832
L 0.570L= - 0.356X0.910 L=-0.568
S S=0.568X0.775~+0.440
eJ eJ=0.356X0.404-0.568X0.268= -0.008
Jg 0.650 Jg= 0.008 Jg= +0.012
n n= - 0.012X0.752= -0.009
gh gh= -0.012 X0.650= -0.008
Compvla//ono/ .Ps
p
·E 0.910 E = - 0.796 E= - 0.875
p p= -0.143
ij ij= 0.583+0.875X0.404= +0.937
F - 0.596X 1.133 F- 0.404 F= -0.705 F= +0.653 F
X X=-l.133X0.653=-0 .. 740 X
0.803 X+0.910 F= O X = - 1.133 F
0.404 F+0.596Xl.133 F=0.937 F=+0.868
X= -l.133X0.868= - 0,983
· comjJulallon of" P.f?
n o
D 0.910 D= 0:324 D= + 0.356
S' 8'=-0.745
e'J' e'J'= - 0.356 X 0.404-0.402= -0.546
J' g' o.650 f' g' = o .. 546 J' g' = + o.840
n' n'= - 0.840 X 0.752= - 0.632
g'h' g'h'= -0.650X 0.840= -0.546
h'i' 0.665 h'i'=0.546 h'i'=+0.821
0' 0'= -0.632-0.821X 0.748= -1.246
t' t'= -0.745+0.840X0.752= - 0.113
0.810 U + 0.885 T + 0.324= 0 0.915
- 0.366
U+T=
Compufahon of P-y
E 0.910 E = -0.530 E = - 0.582
ij ij=0.582X0.404-0.848= - 0.613
X=-1.133 F
F 0.404F+0.596X1.133 F= - 0.613 F= - 0.568
X X=l.133X0.568=+0.644
t t= 0.440+0.009= +0.449
hi 0.665 hi=0.008 hi= +0.012
0 0= -0.009 - 0.012X0.748= -0.018
p p= O= -0.018
u u = + 0.449+ 0.012 x 0. 7 48= + 0.458
0.910 F+0.803 X = O X = -1.133 F
0.404 F=0.665X0.012+0.596 X.
0.450 U-0.448 T+0.402 = 0 1.004 U-T= F 0.596Xl.133 F+0.675 F = 0.008 F= + 0.007
. -0.897 x x = -1.133 x 0.007 = -0.008
Compufaf/o/J o/ PpY
n o P
K 0.570 K + 0.585=0 K = -- 1.026
fg 0.650 fg-- l.026 X 0.268 = 0.155 fg = + 0.662
n n= - 0.6ti2 X 0.752 - 0.805+ l.026X 0.775= - 0.508
l = +0.662X 0.752= + 0.498
gh gh= - 0.662X0.650= - 0.430
hi 0.665 hi= 0.430 hi= + 0.647
0 0 = - 0.p08 - 0.647X 0.748= - 0.99'.l
p p= O= -0.992
U U= + 0.498+0.647X0.748= + 0.982
0.803 X+0.910 F = O X = -1.133 F
o.596 x + o.647 X 0.665 = 0.404 F
F - 0.596X 1.133 F-0.404 F= - 0.430
F= +0.399
x x = - 1.133 x 0.399 = - 0.452
Co/71,,ou/aT/Or? o/ Pz
n o p
5
C 0.910 Ci= 0.415 C= +0.456
K 0.570 K = - 0.456X 0.910 K = -0.72S
fg 0.650 fg + 0.456 x 0.404 - 0.728 x 0.268 = 0 fg=
+0.017
n n - 0.728 X 0.775+0.017 X 0.752 = 0 n=+'."l.551
gh gh= - 0.017X 0.050= - 0.011
t t= 0.017 X 0.752·=+0.013
hi 0.665 hi= 0.011 h'/,= + 0.017
0 0 = 0.551 - 0.017X0.748= + 0.538
p p= 0 =+0.538
U U= 0.013+0.017 X 0.748= + 0.026
0.803 X+0.910 F = O X = -1.133 F
0.596 X - 0.404 F= - 0.017X 0.665
F - 0.596X 1.133 F - 0.404 F= - 0.011 Ji'= + 0.010
X X = - l.133X 0.010= - 0.011
f' g' 0.650 f' g' = 0.043 + 0.456X 0.404 j' g'=+0.349
n' n'+0.918 + 0.349X 0.752 = 0 n' = - 1.180
g' h' g' h' + 0.349 X 0.650= 0 g' h' = -0.227
t' t'=0.349 X 0.752= + 0:262 .
h' i ' 0.665 h' i' - 0.227 = 0 h' i' = + 0.341
O' O' = - 1.180 - 0.341 X 0.748= - 1.435
0.810 U+ 0.885 T + 0.415= 0 U+ l.093
T + 0.512= 0
0.450 U+ 0.448 T + 0.043= 0 - U+0.996 T
- 0.096 = 0
T 2.089 T= - 0.416 T= - 0.199
U U= 0.218-0.512= -0.294
V V - 0.199 X 0.210 - 0.294 X 0.208 = 0.918 V =
+ 1.021
TJ\e results of the foregoing analysis a re summarized
in the t ables below, which are self-explanatory.
The open spaces in the t ables could have beeri filled
in with zeros, but this was not thought necessary.
Also some of the tables might have been reduced in
length by omitting members not affected by the
stresses whiah a re recorded in t hat table, but the values
in different tables are combined by mult iplication,
and that proc.ess can be done more easily when t he
tables are placed side by side and the same· member
appears in the same posit ion in each t able.
Member
c ______ __ ______ _
n _____ _________ _
E_ ___ __ _ , _______ _
F ____ ______ _____ _
K----- --•-------
Lw- _-_-_-_-_~__-_-_-_--_-_-_-_-_
p ___ , ___ ____ ____ z ________ __ __ ___ _
R__: ____ _____ ___ _ s ___ ____________ _
T,, _ ___ __ ________ __ ___ _____ ____ __ _
v _ ____ __ ___ ___ _
x ___ __ ___ __ ____ _
,y, _ ___ __ __ __ ___ __ ______ __ __ __ ___ _
fg ______ __ _____ __ hghi_ _ ____- -_-__- -_-__-_--_-__- _- -_
ij __ ____ ____ _____ _
n ____ __ ___ _____ _ _
o_ - - --- --- ------­P
---- -- -- - - - ---- ­s
--- --- --- ------ ­,
u_ ____-_ _-_-_--_-_ -_-__-_"-__- -_-_
!e'' !g''- _-__--__--__--_-_-__--__- -_
g' h' - - --- - ---- ---
';;, ~'--:============
~:================ t'--- ---- --- ------
6
TABLE 4
p Pw p,. P s
- 1, 736 ___ __ _____ ____ __ +o. 45t __ ___ ___ _____ __ _______ _ + 3 · ~~~ : : : : : : :: :::: : : : : :: :::::: -~~--~~ - :..:o~ii7Z -:..:o~5iij
- 1, 752 +o. 3~ + o. 653 +o. ou + o. oo. + o. 86! -o. 56l;
t ~: ~~g-~~~: :=~~~~~ :~~=~~ :=~~~~ :::::::: :::::::: oo ____________ ____ __+__1._ 0_0__0 --+-1-.- o-o-o- -_-__-_-_-_- -_-__-_-_-_--_-__--_-__-_-_-_- -__- _-
g :::::::: :::::::: :::::::: -~~--~ -+i.-ooo :::::::: o _____ ___ ______ __ -o. HK +o. 236 ______ __ _____ __ 0o -_-_-__-_--_-_-__--_-__-_-_-_--_ +-01.. 022914 +-0o.. 863582 -__-_- -__--_-_-__--_-__--__-_- -_
+ 2, 848 -o. 45' - o. 740 - o. 011 - o. oos -o. 983 + o. 644
o ___ ____________________________ _ _______ _ + 1.ooc
+ 650 - - - ----- -0. 705 __ · __ ---- - 0. 008 -- ----- - - -- -- - - -
- 3, 214 +o. 66l +1. ~ +o. 011 + o. 012 _____________ __ _
+ 2, 128 -0. 43( -0. '105 - 0. 011 -0. 008 ------- - - ----- --
- 3, 200 + o. 647 +1. ooo + o. 011 + o. 012 ______ __ ___ __ __ _
211 ____ _____ _________ ______ ____ ___ _ +o. 937 -O. fi13
+ 490 -0.W -0. 8H + 0.551 - 0. 009 ___ ___ __ ___ ____ _
+ 2, 951 :-o.99; -1.60! +o. 538 - 0.018 _______ _ _______ _
+ 2, 951 -o. 99; -1. 609 + o. 538 - o. 018 - 0.143 ___ __ __ _
-12, 257 ----- --- -0. 009 - - -- - -- - + 0. 440 __ __ ___ _ --------
-14, 719 +o. 49l +o. 807 +o. 013 + o. 449 ____ __ __ ____ __ _ -17,,180 +o. 98 +i. 600 +o. 026 + o. 458 ___ __ ___ ___ __ __ _
+ 358 ____ ___ _ - - - --- - - -- -- - -- - -0. 546 - -- - - - - - - -- - - -- - - 2, 080 ___ ____ _ _____ ___ +o. 341 + o. 840 ___ _____ __ ___ __ _
+- 21,, 315022 -__--__--_-__-_- -_-__- -_-_-_ -__- -+0o.. 324217 +-0o.. 854216 -_-_-__-_-_-_--_ -__--__-_-_-_- -_
+ 5, 566 - - - --- -- --- - ---- -1.18( . -0. 632 - - - -- -- - -- - - - --- +i: 1~ :::::::= ::::::::-~~~~~ =~: m::: ::::: ::::::::
Member I PoPwQ PoPnQ PoPsQ
Q I PwQ
6. 63 - ---- - --- - - ------- -- + 3. 023 -- -------- -------- --- - ----
8. 19 - --- ------ --- ----- -- - - ------ -- + 2. 9lfj -- -- ---- - - ---- --
8 .. 64 - - -- -- - - -- -------- - - -- -- --- - -- - - --- ----- - 7.560- 5. 028
9. 65 · + 3.850 + 6. 301 + 0. 097 + 0.06!! + 8. 376- 5.481
~~: ~ _:-:_~=~~ - :..: -85~8.3{ __ :-:~=~~~~ -:..: -52~256 :::::::: ::::::::
~~~: ~ :~~~=~~ :~~~~~~~ ::~;~.:~ :::::::::: :::::::: :::::::: 78. 00 --- --- -- -- --- ---- - - - --- -- -- --- + 78. 000 ----- -- - -- ------
9~: ~ :::::::::: :::::::::: --=Tf:-~r-n~ ~~~=~ :::::::: 8. 32 ------ - -- - - - - - -- - - - - + 8. 495 + 6. 922 - --- ---- -- -- - ---
61. 15 - 27. 640 - 45. 251 - o. 673 - 0. 489 - 60. 110 +39. 381
92. 66 _____ ____ _ -- - -- "---- -- - ----- -- ---------- --- --- - - +92.660
10. 35 -- -- -- - --- - 7. 297 --- ------ - 0. 083 ---- --- - ---- - - --
132. 80 + 87. 914 +144. 088 + 2. 258 + I. 594 -- - - -- -- - -- -- -- -
10. 35 - 4. 451 - 7. 297 - 0.114 - 0. 083 - - --- -- - ---- - ---
130. 50 + 84. 434 +138. 33( + 2. 219 + I. 56li --- -- --- -- - - ----
6. 95 ------ - -- - - - ------- - ---- -- - --- - ----- - -- - + 6. 512 - 4. 260
-6. 03 - 3. 063 - 4. 92( + 3. 323 - 0. 054 --- ----- --·-----
5. 86 - 5.813 9.429 + 3. 153 0.105 ____ __ __ ______ _ 5. 30, - 5.258 8. 521' + 2.851 0.095 --0.758 ___ ____ _
7_05 __ __ __ ____ 0.oo:i - ---- ----- + · 3. 102 _, __ ___ _ - - -- - ---
6.87 + 3.421 + 5. 544 + 0.089 + 3.085 - - -- --- - -- -- -- - -
6. 20 + 6. 088 + 9. 92( + 0. 161 + 2. 840 - --- ---- --- --- - -
10. 35 -- -- - -- - - - - ----- --- - -- - ------ - 5. 651 ---- --- - - -- - -- - -
132. 80 - ----- - --- - - - - - --- -- +46. 347 + 111. 552 - -- - - -- - - - - - -- --
10. 35 -- -- - - ---- -- ------- - - 2. 349 - 5. 651 -- -- -- -- - - -- - ---
130. 50 - - - - -- - --- -- -- ----- - +44. 501 +107.141 - -- - --- - - "----- -
6. 03 -- - -- - - -- - - - - - ----- - - 7.115 3. 811 -- - - ---- ----- -- -
~: ~1::::::::~: :::::::::: --~-~~= 8: ~~ :::::::: ::::::::
----- 1----- 1--- - - ----·--------------------- --- - --------f---- c ____________ ________________ __ ___ ___ - 5, 248 ____ __ ________________ _____ _____ ____ ______________ __ __ __ _____ _ - - ---- - --~ ____ _______ ________ _
D _______ ______ --- -- --- --- - --- -- --- -- - - --- - - --- -- - 9, 538 __________ - - - - --- - -- ---- -- ---- -- -- ---- - - -- - - --- - -- -- --- -- - -- ---------- - - ------- -
E _______ ___ ___ -- -- -- - --- - - -- -- -- - - -- - - - -- - - --- -- - - -- - - -- - - - - - 3,001 - 1,996 --- -- - ---- --- - ------ -- - -- - - - - - -- -- -- ---- ---- -- - - -- - - - - ----- -
F _____ _____ ___ - 6, 745 - 11, 039 - 170 - 119 - 14, 675 + 9, 603 + 1. 536 + 2. 514 + O. 039 +o. 027 +3. 342 - 2.187 K----- --- ---- - 269, 471 ____ ____ ____ -191, 204 ___ ____ __ ___ ______________ ______ + 71. 056 ___ ____ ___ +so. 418 _____ ____ __ __ ___ __ ___ _____ ___ _
.· L ______ ___ ____ - - - -- --- - --- - 631, 410 - - - - --- - - - - 384, 395 ------ --- - ---- --- - - - ___ ___ __ __ - ---- - ---- ----- --- - - -- - - --- -- - ------- - -- -- --- -- - --
W ____ ____ ____ -- - -- --- - -- - -- - ----- - - - - - - --- -- -- - ------------ --- ------- --------- - +134. 500 - - ---- -- -- - - --- --- - - -- --- -- - -- --- -- ----- --- - - - - ---
p _____ ________ --- ~--- - --- ------------ ----- -- -- - --- - --- --- -- --- - -- --- - -- ------ -- - --- ----- - --- ------ - - --- - --- -- -- - -- - --- - - - ---- -- - - - --- - -- --- z ______ __ ____ --------- --- ---- -- ----- - ---- --- --- --- --- --- --- ---------- --- --- ---- ---- ------ --------- - ---------- ---------- --- --- ---- --- -------
R , ___ ___ __ ____ ----- -- ---- - -- - - -- -- ---- ----- -- --- - -- -- -- -- --- ---------- --- ---- --- - -- ------ - - ------- - - ---- ----- - ----- -- --- - -------- - - -- --- ---- s _____ ___ ____ ------------ ____ __ , ___ __ ----- --- -- -- -- --- ----- ---------- --- --- ---- ---- ----- · ---------- ---- --- --- ------ ---- -------- -- ----- --- --
T _______ __ ____ -- - - -- -- -- -- - -- ----- - -- - -- - -- -- - - - -- - - --- - --- - ------- - -- --- --- ---- -- -------- ------- - -- - -- - - --- -- -- -- --- -- - ___ : __ ____ - -- - - - ---- u _ __________ ----- ------- -- ----- -- -- - ----- ----- ---- --- ----- ------- --- --- ------- -- -------- ------- --- ---------- -- ----- --- ----- --- -- --- --- ----
v _____ __ _____ ---- -- ---- -- ---- --- --- -- --- ------- ---- --- ----- ---- ------ ---- ------ -- ------- - -------- -- --- --- ---- ------- --- --- -- ----- x __ ______ ____ - 78,719 - 128,875 - 1,911 - 1,393 - 111, 193 +112,1.51 + 12,493 + 20.453 + o.304 +0.221 +21. 110 -11. soo
y __________ __ _ __ _____ , ____ ___ ______ ____ ______ ____________ ___ ______ ____ , __ ___ ____ _____ ___ ___ ____ , ______ ___ ______ __ __ __ __ _ ___ __ __ ___ ____ ___ __ _
ef ____ ___ ______ -- - - - ------- 4, 743 ---------- 54 __________ __ ____ ______ __ ______ --- ----- - - ______ ______ _______ _ _______ ___ - - - - -- -- --
fg __ ___ __ __ ____ - · 287,830 - 471, 744 - 7, 393 - 5,219 . ~ --- --- -- - --- --- - - - + 58,199 + 95.386 + 1.495 +I.055 ___ ____ __ _ - - -- - ---- -
gh ____ ________ - 9,472 - 15, 528 - 243 - 177 ----- - ---- - ----- - - -- + 1.914 + 3.138 + 0.049 +0.036 --- ---- --- ----------
hi__ ___ ____ ____ - 277, 788 - . 455, 106 - 7, 301 - 5, 152 --- - - - - - -- - -- ----- - - + 54. 629 + 89. 500 + 1. 436 +L 013 - - - ----- -- --- -- - - - --
~:: : ::: ::: ::-:: -:..: --·-1;5oi -:..: ---Tiii -+--i;ii2ii -:..: -- ---- -:it _:-: __ ~~=~~ -~- - - -~~ -+--i~55ii -+--2~-1w --:..:-i.-iiiiii - - -+0~021 :::::::::: ::::::::::
Q _ __ __ _______ _ - 17, 154 27,825 + 9,305 31( __ ____ ___ _ __ __ ___ ___ + 5, 766 + 9.354 - 3.128 +0. 104 _________ ______ ____ _
P ------------ - - 15,516 25, 166 + 8, 413 28( - 2, 237- ___ ___ ___ + 5,216 + 8.460 - 2.828 +o.094 +0.752- - - - ------
L ::::::::::: -:..: ---50;3.54 :'.: 81. ~ -:..:--i;sio ~~: m :::::::::: :::::::::: -+--1.·104 -+--2:161 --+-0~0« -- -+1~53e :::::::::: ::::::::::
u ________ _____ - 104, 592 - 170,426 - 2, 766 48, 791_ ___ _______ ___ ____ __ + 5. 978 + 9. 741 + 0.158 +,2. 789 ---------- - ---- - - - --
f" < • Hm ~m•••••••••• • • - • = - -•••••• =:. L::::::::::: :::::::::::: :::::::::::: ------- -91 + 1
•
9~t :::::::::: :::::::::: ::::::::: :::::::::: :::::::::: :::::::::: :::::::::: ::::::::::
l; _ ___ _____ ___ - 1,119,142 ~2,025, 103 -491, 047 -1, 087, 001 -192,511 +120,688 +354. 547 + 243.800 +46. 299 +6.90 31. 264 -19. 987
7
TABLE 4-Continued
Member
SOLUTION OF WORI( EQUATIONS
Before substituting in the work equations the numer­ical
values just obtained, a few suggestions are offered
regarding the arrangement of the variables, the system
of tabulating, the elimination of the variables bv a
method of division and addition and the order in which
the various equations are combined.
The position of the variables iu the equations is of
great importance since their proper arrangement means
the saving of considerable time and labor. When one
·Of two redundant cross wires will be in action, the one
selected as the more probable should be placed at the
end of the equations, should it then be found after
solving the equations, that the wrong wire was selected,
only the last term of n- 1 equations will have to be
·changed and the ;nth equation will have to be rewritten
as it is composed entirely of terms involving the wire
which is to be interposed.
When by observation it appears likely that the
resultant stress in a wire will be compressio1;, thereby
putting it out of action, that wire should be placed
either in the first or last term of each equation, and the
equation made up entirely of functions of the stress in
that wire should be either the first or last of the series
of simultaneous equations. Should it so happen that
this equation would appear in the middle of the series
of equations, there would be considerably more work
involved when it is found necessary to leave out this
equation, as it has been combined in the process of
solving simultaneously with the equations preceding
and following it.
When it is found that a member chosen as redundant
in a structure, will go out of action, the number of
simultaneous equations is reduced by one. When by
fnspection it appears that two or more members
chosen as redundant, may go out of action, the equa~
tions involvipg the stresses in these members should be
placed first and last in the series of equations and the
terms in the remaining equations which involve the
stresses in these doubtful members shonld be placed at
the beginning and the end of the equations in which
they appear.
In short, always place the equations involving the
stresses in redundant members, most likely to carry
the stresses in a structure, in the middle of the series of
equations to be solved simultaneously.
The system of tabulation as previously introduced, is
follo_wed in the writ ing and solving of the work equa­tions.
The solution of the equations simultaneously
is effected by dividing each equation by the coefficient
of th,e first variable in that equation. The equations
are designated 'by numbers which appear in the left­hand
margin of each page. Before dividing the various
series of equations by the coefficients of the first
variable, it is suggested that the signs for that series
be placed in their respective ·positions in the table so
that the signs of the first ter~s of the equations ~ill
alternate. The sum of these terms will then be equal
to · zero and the number of variables in the next series
of equations will be reduced by one.
The combination of adjacent equations by addition
is necessary because of the relative importance of the
position of each· equation in the series. The order in
which the various equations are combined, is indicated
in a column at the lef-t of the page, for example, equa­tion
7is obtained by adding 1 and 2, 8=2+3, etc.
8
The solution of the work equations now follows:
T ABLE 5
Xw
l_ _ ___ - - -- - ·-- - - - -- - - - --- --- - --- - - - - -- - ------ - - - - - - - - - - - +354. 547
2 __ ____ __ __ - - - ---- - - - -- -- -- - --- ------ --- --- -- - - - --- - - -- + 243. 806 3__ __ _____ ___ _____ __ __ ________ ____ ___ _____ ___ ______ __ __ 46. 299
4 ___ _ - - - - - - ------ - - ~ --- - 5 __ _____ __ _._ _ : __ __ ___ ____-_-_--_-_-_-__--__--__-_-_-_-_- -_-_-_-__- -_-__-_--_-__- -_-__- ++ 316.. 92f0i42
6 _____ _____ __ , ____ - -·- ---- -- - - -- --- -- - -- - - - - -·- - - - - --- - - - - 19. 987
!_ __ __ ______ ___ ______ ______ _______ ____ ______ __ ___ __ ___ _
32 ___- _-__-_- _-_- -_-_-__-_-_- -__-_-_--_-__-_-_-_-_- -_-_-__-_- -_-_-__--_________ :_ ___- _- -_-__-_-_- -_-_-__-_- -_
4 __ , __ ___ ____ _____ ___ ___ ______ __ ___ ___ ________ _____ __ _
56 .__._. -_-_-_-_-_-_-__-_- -, -_-_-__--_-_~__--_-_-_ -_-_-__-_-_-_-_--_-_-_-_-__-_--_-_-__--_-_-_-__--_-__- _--_-_
7 ____ ___ __ ___ __ ____ ________ _________ _______ ____ _______ _
8 ___ ___ ___ __ __ ____ _____ ___ __ ____ ______ ___ _______ ______ _
9, ______ __ _______ __ __________ , _______ ____ ___ __________ _
1I10 -_-__-_--_-__-_--_-_-__-_-_-_-_--_·_-_-_- -_-_-_-__-_--__-_-_-_--__-_-_--_-_-__- -__- -__- -_-_-_-__-_-_- -_-_-•
78 . ._ _-_-_-_--__--_-__- -_-_-__-_-_--__- -__- -_-__-_--__- -_-__--_-__-_-_--_-__- -_-_-_-__- -_-_-_-_-__- -__-_-_-_- -- -------- ---------- ----
9 _____ ____ ____ __ ______ , ______ ___ __ __ ______ _____ _______ -- --- ----- --
10 ______ ___ _____ __ _________ ___ __ ___ __ _______ __ _________ -- - - -- --- - --
+243. 806
+632. 294
- 6. 599
+ 60. 040
+ 51.171
- 32. 7~1
+ 0.688
2. 594
0. 143
8. 700
+ 1. 637
1. 637
I. 906
2. 737
8. 843
7. 063
0
- 1
+1
- 1
+1
lL ______ ____ _____ ___________ ____ ________ __ __ ____ ______ --- - - - -- --- - ----- - ---- --
12. - - - - - - -- --- ----- - - - --- - - - - --- - ---- --- -- -- - - --- ----- - 7+8 --- -- - - ---- -
13 __ ___ ___ __ ___ ___ __ __ ____ --- -- -- - --- ---- - - --- -- -- --- - - 8+ 9 -- -- -- -- - -- -
14 ___ _______ - - - -- -- - -- -- ---- --- - -- - - --- - -- - -- - -- - -- -- -- 9+ 10 - --- -- ------
Xz
+ 46'299
- 6. 599
+174. 826
+ 99. 089 + o. 337
0. 488
+ 0.131
+ 0.027 + 3. 776
- 14. 357 + 0. 011
0. 024
+ 0. 158 + 3. 803
_;_ 10. 581
- 14. 346
0. 013
+ 0. 083
1. 390
1. 197
+ 2. 031
0. 013
1. 307
2. 587
+ 0. 834
U::::::::::::::::~:::::::::::'.::::::::::::::::::::::::11::::::: ::::: ::::=::::::: ~~
ii;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;::-----iiti~ -:;; ;: :; ;; ;;: : :: : : -:l :
17 --- - - ---- -- - - - - --- - - --- - -- ----- - - - --- - -- - - - - - -- - ~ - --- 13+ 14 - - - -- - ---- -- - - ----- - - - - -
Xa
+ 6.902 + 60. 040
+ 99. 089
+323. 819
+ 0. 554
0. 353
+ 0. 019
0. 246
+ 2.140
- 46. 917
+ 0. 018
0. 018
0. 227
+ l.894
- 44. 777
- 46. 899
0
0.119
0. 692
5. 064
+ 6. 640
0
0. 811
5. 756
+ 1. 576
0
+ 0. 621
2. 225
+ 1. 890
+ 0. 621
1. 604
0. 335
15 _____ ___ ____ ______ ______ __ , __ ___ _______ _____ ____ __ ___ --- - - - - - - - - - --- - -------- --- - - - - -- - -- ·+1
16 __ __ - - ------ - -- - - __ , ____ _____ , __ __ ___ ______ . ___ ---- - - - - - -- --- - - - -- - - - - --- - ---- - ___ , ______ - - 1
17 ____ -- ----------- - -- ___ __ : ___ , __ __ -- - -- - --- -- - - - - - - - - - ----- - - - - - - - - - - --- - - - - - - - --- - - -- - - - + 1
1198 _-_- ,- _--__-_-_-_--_.-_ _-_- :- -_-_-_-__--__-_--_-_-_-__- -_-_-__-_-_- .-_ _--_-_-__-_-_--_-_-__--_-__--_-_-__- -_ 15+16 - - - -- -- - - --- ------------ - - - -- - -- - - - - 16+ 17 ------ ' ----- - -- - --- ----- --- - - ---- - --
18 __ ________ ____ ____________ __________ __ --- - - ------- - - - - - - - - -- - - - - ---- - ---- -- - - - - - --- - - - - --- - - ---- -- -
19 __ ______ _________ ______ ______ _______ ___ _____ ____ ____ --- --- -- - -- -- -- ------- - ---------- -- -------- - ---
x.
+ 31.2fi4
+ 51.171
+ 0. 337
+ 0. 554
+174. 683
- 43. 062
+ 0.088
0. 210
+ 0. 007
0.080
+ 5. 587
2. 155
0.122 o. 20.1 o. 073
+ 5. 507
+ 3. 432
0.064
+ o. 074
0.008
0. 780
+ 3. 432
+ 0. 010
+ 0.066 o. 788
+2fi4. 000
0.008
+ 0. 026
- 0. 945
+263. 992
+ 0. 018
- 0.919
+425.108
+ o. 011 I
+ 2. 743
+ 425. 119
+ 2. 754
+1
-1
Xy
- 19. 987 l, 119, 142
- 32. 721 2, 025, 103
- 0. 488 491, 047
- 0. 353 l, 087, 007
- 43. 062 192, 519
+126. 671 120, 688
·O. 056 + 3, 156, 541
+ 0.1 34 8, 306, 207
0. 011 + l 0, 605, 996
+ 0. 051 -157, 491, 596
- I. 377 + 6, 157, 849
+ 6. 338 6, 038, 325
+ 0. 078 5, 149, 666
+ 0. 123 + 2, 299, 789
+ 0.040 -146, 885, 600
- I. 326 -151, 333, 747
+ 4. 961 + 119, 524
+ 0. 041 - 2, 701, 819
0. 045 840, 259
+ 0. 005 - 16, 610, 381
+ 0.188 + 21, 426, 270
+ 4. 961 + 119, 524
0.004 - 3, 542, 078
0. 050 - 17, 450, 630
+ 0.183 + 4, 815, 889
+381. 615 + 9, 194, 154
+ 0. 003 + 2, 710, 083
- 0.019 6, 745, 508
+ 0. 219 + 5, 774, 447
+381. 618 + 11, 904, 237
- 0. 016 - 4, 035, 425
+ 0. 200 971, 061
+614.•522 + 19, 169,4fi4
- 0. 010 - 2, 515, 851
- o. 597 + 2, 898, 690
+614. 512 + 16, 653, 613 o. 607 + 382, 839
+ 1. 446 1 + 39, 174
+ 0. 220 139, 012
+L666 Xy= - 99.838
Xy=-59.926
Since " Y" is a wire and the min us sign signifies com­pression,
the results show " Y " is out of action. There­fore,
in the above equations all functions of Xy are
left out and tttie resulting equations solved. Here the
value of the arrangement of terms can be seen. <.The
value of X 8 is obtained from equation 19 without any
more work being necessary. The reason for not using
18 can bs seen by going back to 6 which is mad:e up
entirely af "Py" terms. 11 is the s'um pf 5 arid 6;
15 of il .and 12, and 18 of 15 and 16, so 18 i~ dependent
upon "Pcv" which is no longer to be considered.
From 14 Xz = 5774-4758+131 = +1148 lbs.
The completion of the solution is as follows:
From 19 X8 = + 139 lbs·. .
17 XR= 2899-2.743 X 139= +2517 lbs.
!.l
10 Xr = 21426 - 2332 - 16715+108= +
2488 lbs.
5 Xw = 6158 - 4072 - 13-45-777 = +
1251 lbs.
These values show the wires chosen as redundant,
excepting Y, are in action. The values are then sub­stituted
in the following table and the total stress in
the various members obtained by taking the algebraic
sum'1of ail 'the stresses in each member.
When the resultant sign of the stress in a wire is
negative that wire is out of action and the whole
analysis must be done over leaving that wire out of
the structure.
9
TABLE 6.-Final summation of stresses
Member Po PwXw PzXz PaXa
c_______________ __________ ____ ____________ ________________ - 1736 _____ ___ ____ _____ _______ + 523 ________ ____ _________ __ _
D---- - --------- - ------------------ - --- - - - - - - - - ------------ - 3271 - - - - - --- - - - - - - -- - - - E______ ___ _______________________ ____ _____ ___ ______ _______ - -- - - - - -------- - - + 896 --- - --- - - - -- + 397 ________ ___ _ __ ___ ___ ____ - --- - -- - ---- - ---- -- - - --- -122
F_ -- ---- --- - --- ------------- -- ------ - - - - ----- - - - - - ---- -- - - - 1752 + 499 + 1625 + 11 + 18 +l~l
K-------------------------------- ------------ ------------- + 3891 -1284 --- --- - - - - - - - 836 --- -- -- -- - - - - -- - - -------
{v_~~: : :: :::: : :: :::: : ::::::: :::::: :::::::: :: :: ::::::::::::: + 735& ---- -+1251- _____ :-:::'.21
_ : ::::::::::: ___ __ :-::~~?- ::::::::::::1
zp_ _ -__--_-__-_-_-_--_-_- - -------- -- ------ - -- - -- ---"-- ------ - - ------ -- ._ __ _____ ---- -- -- ----- -- - + 2488 - - ------ - - -- - ------- --- - --- - --- - -- - - - --- -- ~------ - - - ----- - -- ---- ----- --- ---------- ___ _________ __________ __ ___ ______ _._ _ +n48 _____ ____ __ _ ______ ___ , __
R_ -- ------- -- ----- -- -;------------ ----- --- -C-- ----------,- --- --------- _ _________ __ __ __ _____ __ _ ______ _____ +2517 ------ -- -,- -
8 _____ - - - -- - - -- - - ----- - - -- - - -- - --- - - - - - - - - - -- - - -- --- -- - - - - - --- - - - - - - - -- - - - - --- - - - -- - -- - - --- - - - - - - -- ---- ---- - - - -- --- - -- - +139
TU- -_-_-__- -_-_-_-_-_-_-_-_-_-_-__'_-_-_--_-_-_-_-_-_-__-_-_--__-_-_-_-_--_-_-__-_- -__- -_-__--_~__--_-_-_-_-_-_-_-_ ------------ ------------ ---- -------------------- ------ -------- -- ---- ---- · -- 323288 +-1 655964 -_-_-__- -c- -__-_-_-_- -_
V ___ _______ __________ ________ _ , ___ ______ __ ________ , ______ --- - - - --- --- - ---- - - - - - -- - - - - - -- - -- -- -1172 +2094 - - - --- - ---- -
X ________ _______ _____ , ______ __ ______ ___ _______ ___________ + 2848 - 565 - 1841 + 13 - 20 -137
y- - - -- -------------- - - - -------- - - --~- - - - - -- - --- - - - - --- - -- 0 - - - -- - --- -- - -- - - - - - - - - - - - - - - - - - - - - - - -- -- - - - - - -- - - - - ----- - ---
ef _____ - --- - ---------- -~---- ----- -------- - - -- - - - - - ----- - -- - + 650 ____ __ _____ _ - 1754 ______ _____ _ 20 --------- - --
fY- -- - -- - - ---- - --- - - - - - --- - --- - - - - - - - - - - - - - - --- - - ---------- - 3274 + 828 +2699 + 20 + 30 ---- - -- --- --
gh. ------------ ------ -- ------- ----- --- -- ---------- --------- + 2128 - 538 -1754 - 13 - 20 ---- - -- - - ---
hi__ ___ ______ ________ ___ ___ ______ ______ ____ __ ____ ___ _______ - 3290 + 809 +2637 + 20 + 30 ----- - - - - --- .
ij________________ ___________ ___ ___ ____ ____ ____ ____ _________ - 217 - ---- - - - - -- - -- -- --- --- -- -------- - -- - -- ------- - - - +130
0n_-_-_-_-_--_-_-_-_-__-_-_-_-_-_-_-_-_-_- -_-_-_-__--_-_-_-_ -_-_-_-_-_-_-__--_-_-_-_-_-_-_-_ -__-_--_-_-__--_-_-_-_-_-_ ++ 2945910 -- 1623416 --24003003. ++ 661383 -- 4253 ---------- ---------- ----
t::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~l~~i _____ :-:::~=- =400J -----~-~=~- :;:1!6~ ______ :-:_~_ !___ _____ ______ ____ ___ ___ __ __ ___ _____ ___ __________ ____ ___ __ -14719 + 623 +2008 + 15 + 1130 -- -- - - --- - --
u____________________ ______ ___ __ _______ __ ________ ___ _______ -17180 + 1228 +3981 + 30 +1153 -- - ---------
·e'f'- ------ - ---------------------- - ----- - ----- ----- - - - --- --- + 358 - - - -- - -- - -- - - - --- --- -- - - -- - ----- - - -- - 1374 ------------
f'g'- --- --- -- ------ -- ----- ------·----------- -- --------------- - 2080 ---- -- -- - - - - - --- - - ------ + 401 +2114 ------------
g'h'-- --- ---- ------,-- -- ----- --- -- ------- ----- -- ---- ------ -- - + 1352 -- - ------- - - - --- - - - - - --- - 261 -1374 - - ----------
h'i'- -- --- - - - --- - --- - --- - - - - - -- - - - - - --- - --- - --- - ------ ------ - 2102 --- --- - - --- - - - -- ---- - --- + 391 +2066 - -- ---------
n' -- ----- ---------- ----' -- ---------- -- --- -- ------ ----- ----- + 5566 - -- - ------ - - - - - --- - - - - -- - 1355 -1591 - - ----------
0' -- - ------ - - -- - - ---- - - - - -----·- - --- - -- -- - - -- - - - --- -- - "----- + 7138 - - - --- - - ---- - - - --- - - - - -- -1647 - 3136 - --·---------
•' - - - ---------- - ------- - ------- ---- - - - - - - - - -- --- - - - - -- - ---- + 1510 : ::: : :: ::::: : :::::: :: : :: ______+ _ _ 301 _! -1s15 -- ---·------
,, - -- - - -- - -- - - -- - - - -- - - - ---------- ----- - -- --- - - -- - - - - - - - - - - - 54 - 284 - -----------
Total
- 1213
- 2375
+ 275
+ 522 + 1771 + 3605
+ 1251 + 2488
+ 1148
+ 2517
+ 139 + 366
- 1994 + 3266
+ 272
0
- 1124 + 303
- 197 + 206
87
- 1566
- 1720
- 1740
-11172
-10943
-10788
- 1016
+ 435
- 283 + 355 + 2620 + 2355
365
37
CONCLUSIONS
The following ou.tline summarizes the procedure and
suggestions detailed in this report:
bers and solve for the stresses due to this load. Tabu­late
these values under the heading "P" with sub­scripts
designating the various redundant members.
1. Show completely dimensioned line drawings of
the framework to be analyzed.
2. Tabulate the physical properties and directional
components of the members in the framework and any
others which will affect the stresses in these members.
3. Make line drawings of the framework, showing
the point of applic.ation and the magnitude and direc­tion
of all applied loads affecting the framework. Give
the nomenclature to be used.
4. Very carefully select the rudundant members try­ing
to choose those which may go out of action.
5. Writ,e the work equations which will be \)qual in
number to the· number of redundant members.
6. Make a line drawing to show the statically deter­minate
structure obtained by temporarily removing
the redundant members and solve this structure for the
stresses due to applied loads. These values" ~re 'ta.bu­lated
under the heading P0 . Show in the right or. left
hand margin of the page, which l)'.lember is being solved
for by the equation on that line.
7. Show by a line drawing that part of the framework
affected by a unit load in each of the redundant mem-
8. Tabulate the foregoing values and solve for the
numerical va lues required in the work equations.
9. Before solving the work equations, arrange the
various terms in the equations so that those represent­ing
members which may be found to be out of acti0n
will appear either at the beginning or the end of the
equations.
10. Arrange the various equations in a series so that
the equations composed entirely of the doubtful mem­bers
will appear at the beginning or end of the series.
11. Tabulate the equations according to some such
method as previously shown and solve simultaneously
by dividing each member in turn by the coefficient of
the first variable in that equation, changing the signs
so that the sum of the first t erms in adjacent pairs of
equations will equal zero. Continue this process until
soluti~n of the equations is complete.
12. Multiply the stresses in the various members,
due to unit loads in .the redundant members, by tμe
actual loads obtained in these members, and take the
algebraic sum of these stresses, together with those
under the heading P0 , and obtain the final stress in each
member of the framework.
10
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