Analysis of elastic arches, three-hinged, two-hinged, and hingeless, of steel, masonry, and reinforced concrete by Balet Joseph W

Author:Balet, Joseph W. [from old catalog]
Language: eng
Format: epub
Tags: Arches. [from old catalog]
Publisher: New York, The Engineering news publishing company; [etc., etc.]
Published: 1908-03-25T05:00:00+00:00

When the analytical method is applied, a table like VII should be arranged (see end of chapter). The first line gives the values of the ordinates of the arch axis at the panel points I, II, etc.

The second line gives the values of the angle which each plane of section makes ^ith the vertical at the panel points. The third line gives the angle of the curve for each panel, and the fourth the length of the curve in each panel.

In the fifth line is given the depth of the arch rib at each panel point, and in the sixth the moment of inertia at each panel point.

(The average moment of inertia=90.7 ft.*)

(b) Graphical Computation of the Horizontal Thrust H. — Vertical Forces. —In the Appendix the following equations define the locus and tangent curves for vertical forces:

/:

— ydx

H f -; (128ii)

r-^2y^dx I

2

/:

"*"2am J

-r-xax

Xi \ ; (129A)

-^2 3^dx

f

_1 1.

2

Xi ?— I (laoii)

/

r^2dj_

Also,

«'-.=&• #('2x«+a5„-i)+^- ;-7^-(2x„+*«+,). (132.1)

Z CTq i m ^ ao ^ m+1

Substituting these latter values in (128ii), (129ii), and (130A) gives

B -^0 ^mVm ^ (j35^)

1= yi ^ . > (136.1)

H m„,v"„+H

ToB

0 "-m- m ■ "P^r^

Further, for the location of the axis DD,

tf2 -^j ^J*" " . .... (138il)

Again,

In Fig. 35 dfn—dQy etc., all panel lengths being equal. In the present example the following modifications of equations (131ii), (132-4), and (133-4) will give siSSciently accurate results:

Vm-jlim^ ....... (131B)

•*/

^m=Y^^' ...••.. (132-B) e"m=^ (133B)

The first value to be computed is j- (see Fig. 35a). For instance,

J to obtain the value of j- at the panel center X, a panel length he

may be made the unit, and the line ad be drawn parallel to the line ce. From similar triangles.

Now, be is the imit, a6=/o, and bc^I/,

The point d has been transferred to d' on the panel center X.

In the same manner the remaining values of •— have been com-

puted, and the upper points on the ordinates are connected by the line d7> the ordinates of which, measured from ADj represent the

values of -^ at the panel points when one panel length is the unit

of measurement.

The values of -^ have been inserted in the seventh line of Table

VII.

The next value to be computed is that of t/2, or the distance of the axis DD from the axis AB (Fig. 35). From equation (138ii),

and from equation (ISSB),

«' m— r •

If the moment of inertia is assumed to be a constant, it will disappear from the equation; and when numerator and denominator are multiplied by do, the numerator of equation (138-4) will simply represent the sum of all the areas = i't/do=*^rea kCB, and the denominator will represent the span AB=Ido=ly or, the value of the fraction will represent the height of a parallelogram which has the same area as the figure ACB.

In the above equation the variation of the moment of inertia appears as a factor in the fraction.

In either case the values of v"„» can be considered as horizontal forces acting in the axis of the arch, and the value of the numerator is equal to the moment of these horizontal forces with reference to the line AB.

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