TEMPERATURE EFFECTS 89

The complete equation * is therefore

y = 1.25® + 1.76722 x2 + .16695 x3 - .10056 x4 + .01639

H + 1 .2 5 +8.8361 +3.1303 -6.2850 +3.2012 8.8826 0.88826

2 + 1 .2 5 +7.0688 +2.0034 -3.2179 +1.3112 8.4155 0.84155

* Since x in th e expression — is reckoned in panels, th e ta n g e n t of th e angle of slope is o b tain ed b y di-i z di-i y

v id in g th e n u m erical v alu e of — b y th e panel length in feet.

dx

23. Temperature Effects. — We have now determined the axial shape of the arch, which fact, in connection with the preliminary

assump-* Considering a straight line as a parabola of the first degree, it is evident that any value of y is the sum of the ordinates of parabolas of the first, second, third, etc., degrees, and the curve is therefore termed the “ multi-parabolic ” curve.

9 0 PR E L IM IN A R Y ARCH D ESIG N

tions already made as regards size and shape of the arch rib or barrel* at various points along its axis, furnishes a basis for the evaluation of all data included under item 4 of Art. 2. It remains only to determine the probable range of temperature and the thermal coefficient for the ma­

terial employed.

The thermal coefficient, c, for both steel and concrete is generally assumed at 0.0000065, although the figures 0.000006, 0.0000066 and 0.000007 are sometimes employed.

The assumptions to make regarding temperature variation are, of course, dependent upon local climatic conditions. This matter is con­

sidered in somewhat greater detail in the Appendix. The standard speci­

fications of the A.A.S.H.O. covering temperature stresses are as follows:

“ In all statically indeterminate structures, provision shall be made for the stresses resulting from the following variations

24. Conclusion and Summary. — The above determination completes the data needed for making a preliminary analysis of an arch bridge.

There are, of course, certain other conditions such as shrinkage in con­

crete, plastic flow, support displacements, etc., which will cause stress.

These conditions should be investigated but not necessarily while the arch is in its preliminary stage. Such features are therefore reserved for discussion later on. (See Chapters VI and IX and also the Appendix.)

In summary, therefore, the preliminary work necessary before the first clastic analysis of an arch bridge can be begun is as follows: certain modifications (see Chapter IV) may be followed.

CONCLUSION A N D SUM M ARY 9 1

15.00 7.50 10" I '6" F illed spandre — B arrel arch

16.67 8.92 10' x8 0'0" 1'7"X 80'0" Filletl spandre — B arrel arch

20.00 10.00 11" 2'1" Filled spandre — B arrel arch

25.08 8.42 40.0 None 12"X 44'0" 2,lJ" X 4 4 /0 " F illed spandre — B arrel arch

27.00 13.50 36.0 2 ® 6'0" i2 " x 3 8 '6 " F illed spandre — B arrel arch

30.00 14.00 11" 2' 1" Filled spandre — B arrel arch

30.00 4.50 21.0 N one 9"X 24'0" T 5"X 24'0" Filled spandre — B arrel arch 30.00 7.00 18.0 N one 1 0 "x 2 0 '0 " 2 '6 " x 2 3 '0 " Filled spandre — B arrel arch 35.50 10.50 76.0 2 @ 12'2" 12"X l03'0" 2,8J"X 103,0 " F illed spandre — B arrel arch 37.75 18.10 48.5 None 9 " x 5 1 '0 " i 'i o " x 5 r o " F illed spandre — B arrel arch 38.00 19.00 20.0 2 ® 3 '6 " 10" x2 2'6" F illed spandre — B arrel arch 39.50 10.90 21.0 None 10"X 24'0" 1'8"X 24'0" Filled spandre — B arrel arch

40.00 11.25 10"X24'0" 1'8"X 24'0" Filled spandre — B arrel arch

40.00 18.40 12" 2 '3 " Filled spandre — B arrel arch

40.00 8.00 20.0 N one I l " x 2 r 4 " 2'0"X 21'4" Filled spandre — B arrel arch

40.00 10.00 16.0 10"X16'0" 1'8"X 16'0" Filled spandre — B arrel arch

40.00 15.00 20.0 N one 10"X23'0" 1'5"X 23'0" Filled spandre — B arrel arch 40.00 7.00 20.0 None 1 0 "x 2 3 '8 " 2,6"X 23,8 " Filled spandre — B arrel arch 40.00 11.00 20.0 N one 12"X 24'2" 3 '6 " x 2 4 /2 " Filled spandre — B arrel arch 42.00 10.50 24.0 None 12"X26'10" 2'0"X 26/10" F illed spandre — B arrel arch

44.40 13.30 12" x2r0" 5'6"X 21'0" Filled spandre — B arrel arch

45.00 8.15 12"X 24'0" 2 '0 " x 2 4 '0 " F illed spandre — B arrel arch

45.00 12.00 26.0 2 @ 5 '0 " 9}"X 43,0 " 1'10"X43,0 " F illed spandre — B arrel arch 45.00 14.00 20.0 2 @ 5'0" 1 2 "x 2 1 '9 " 2'2"X 21/9 " F illed spandre — B arrel arch 45.00 15.00 20.0 2 ® 4 '0 " 8 "X 30'0" 5 '4 " x 3 0 '0 " F illed spandre — B arrel arch 46.00 8.05 30.0 2 ® 5 '0 " 12"X 36'0" 2 '6 " x 3 6 '0 " Filled spandre — B arrel arch 46.00 12.00 36.0 2 ® 6'0" 12"X 4'6" 1'91"X 4'6" Open spandrel — 3 ribs

92 PR E L IM IN A R Y ARCH D E SIG N

50.00 13.00 18.0 Nono 12"X 21'2" 2,0 " x 2 1 /2 " Filled spandrel — B arrel arch 50.00 9.00 23.0 2 @ 5'0" io " x 3 6 '6 " 2'6/'x 3 6 '6 " F illed spandrel — B arrel arch

50.00 10.00 20.0 Nono 1'2"X 21'4" F illed spandrel — B arrel arch

50.00 14.00 20.0 2 ® 5 '0 " n " x 2 r o " 2'5"X 21/9,/ Filled spandrel — B arrel arch 50.00 19.00 27.0 Nono i2 " x 2 9 '8 " 3 '0 " x 2 9 /8// F illed spandrel — B arrel arch

50.00 10.00 16.0 12"X 16'0" 2/0//X l6 ,0// F illed spandrel — B arrel arch

50.00 10.00 21.0 Nono 12 "x 2 4 '0 " 2'6,/x 2 4 '0 " Filled spandrel — B arrel arch

60.00 11.59 40.0 N one 10//X44,0,/ 1/8"X 4 4 '0 " Filled spandrel — B arrel arch 60.00 11.92 20.0 Nono 1 2 'x 2 3 '0 " 2 '6 " x 2 3 '0 '/ F illed spandrel — B arrel arch

60.00 6.42 16.0 1'3"X 16'0" 2 '6 '/ X l6 ,0 '/ Filled spandrel — B arrel arch

60.00 12.00 16.0 1'3"X 16'0" 2 '8 "X 16'0" Filled spandrel — B arrel arch 60.00 13.66 42.0 2 @ lO'O" i'4 " x 6 6 '0 " 5'8 "X 6 8 '0 "

60.00 10.50 l'4 " x 2 4 '0 " 2'6,/X24'0" Filled spandrel — B arrel arch

60.00 7.37 21.0 None 1'3"X 23'8" 2/6//x 2 3 '8 " Filled spandrel — B arrel arch 60.00 13.80 30.0 I ® 5'0" 1'3"X 35'6" 4/6 i" x 3 5 '6 // Filled spandrel — B arrel arch

70.00 10.00 18.0 N one 1/1//X20'0// 2'9"X 20'0,/ F illed spandrel — B arrel arch

70.00 23.49 2 '0 " x 3 /0,/ 3 '6 " x 3 '0 /# Open spandrel — 2 ribs

CONCLUSION A N D SU M M AR Y 9 3 75.00 11.25 24.0 1'3"X 24'0" 2 '6 " X 24'O'' Filled spandrel — B arrel arch 75.00 11.66 32.0 1 ® 5 '0 " I T 'X 37'2*" 5 '6 " X 3 7 '2 |" Filled spandrel — B arrel arch — 80.00 18.50 20.0 N one 1'4"X 20'6" 1'8"X 20'6" Filled spandrel — B arrel arch 80.00 13.50 40.0 2 © 6'0" 1'9"X 54'4" 2'7 i"X 5 4 '4 " Open sp an d rel — B arrel arch 90.00 14.00 32.0 2 © 6'0" 1'6"X 42'0" 3 '6 "X 42'0" Open spandrel — B arrel arch 90.00 21.00 27.0 N one i '6 " x 5 '0 " 3 '5 " x 5 '0 " Open spandrel — 3 ribs 90.00 15.00 24.0 1 '6 "X 24'0" 3 '0 "X 24'0" F illed spandrel — B arrel arch 91.50 57.75 21.0 N one 2 '6 "X 2 '6 " 4 '3 "X 4 '3 " Open spandrel — 2 ribs 92.47 35.53 27.0 2 @ 5 '0 " i 'i o " x 5 'o " 3 '3 "X 5 '0 " Open spandrel — R ib arch 93.00 26.75 24.0 2 © 5 '0 " 2 '0 "X 5 '0 " 4 '0 "X 5 '0 " Open spandrel — 2 ribs

94.50 18.90 2 '3 "X 3 '0 " 4 '6 "X 3 '0 " Open spandrel — 2 ribs

96.00 28.00 2,8/# 4'10" 3 rib s, 2 @ 3 ' wide, 1 @ 4'6"

100 None 1'9"X 20'0" 4'6' 'X 20'0" F illed spandrel — B arrel arch None 3 '0 "X 4 '0 " 5'3' 'x 4 '0 " Open spandrel — 2 rib s

CONCLUSION A N D SUM M ARY 95 130.00 26.00 18.0 N one 1'6"X 14'0" 3 '0 "X 14'0" F illed spandrel — B arrel arch 130.00 17.00 36.0 2 ® 8 '0 " 2 '3 " x 6 '0 " 5 '7 " x 6 '0 '/ Open spandrel — 4 ribs

96 P R E L IM IN A R Y ARCH D ESIG N

180.00 31.50 21.0 2'9"X 14'0" 4'6"X 14'0" Open spandrel — B arrel arch

182.00 45.50 3 '0 "X 4 '6 " 5 '6 "X 4 '6 " Open spandrel — 2 rib s

CONCLUSION A N D SUM M AR Y 97

SUPPLEMENTARY NOTE

Mathematical A xial Curves for Specified Loadings

(а) The Second D egree Parabola (Uniform Horizontal Loading). — For this case (Fig. 14) we m ay imagine the arch cut at the point a where the axis is horizontal and then write the equation of moments about any

other point 6. Since by hypothesis there is to be no moment a t any point, then H, the only crown force, is in moment equilibrium with the total load wx, from which

TT w x 2

V = ~2~

which is evidently a parabola of the second degree. This is the fundamental equation for the suspension bridge, and the arch axis for this loading is therefore simply the inverted curve of a suspension bridge cable for the same loading. For open spandrel arches having floor system s which are comparatively heavy with respect to the weight of the columns and ribs, this parabola is not far from the true axis and is useful for preliminary layouts.

(б) The Transformed Catenary (Horizontal Fill with Lateral Component of Earth Pressure N eglected).* — The equation of this curve, the derivation of which is not given (as it may be found in any standard text on mechanics) is as follows:

V = j [ c “ + e ~ ] in which, referring to Fig. 15, x and y are the coordinates of any point on the axis with o as origin

y0 = ordinate of axis at the crown e = The Napierian base

The value of a may be found from the above equation by substituting for xi and y i the known coordinates of the skewback through which the axis is to be passed.

The equation of the transformed catenary may also be written y = | [cosh 5 ]

by means of which the axial curve m ay be easily plotted from a table of hyperbolic cosines.

* Lateral component of earth pressure is usually neglected in the design of filled spandrel arches.

Fig. 15.

C H APTER IV

ANALYSIS OF FIXED ARCHES