Two-hinged Arches.—Two hinges are sometimes used at the supports without the crown hinge. Two points upon the line of pressure are thus fixed and the vertical components of the end thrusts may he found by moments about the hinges. As the span of the arch remains unchanged upon the application of the loads, Formula (14) of Section 163 applies to this case, or Let component of the thrust at left support; =moment at crown of all loads bettivecn crown and left support; L `_=half span of the arch axis; h=rise of the arch axis.
Then using the same notation as for the solid arch, Substituting this in (14), we have for arch with vertical loading, The thrusts and moments may now be determined in the same manner as for the solid arch.
173. Unsymmetrical Arches.—The formulas of Art. 16 apply only to arch rings which are symmetrical with respect to the crown sec tion. It is frequently necessary or desirable to construct arches which for topographical reasons are not alike upon the two sides of the crown. In these arches 1 is made constant for the whole arch a division may not come at the crown section, the values of x and y will not be the same upon the two sides of the section nearest the crown, and the formulas produced as in Section 164 become quite complicated.
For this case, the origin of coordinates may be taken at the middle of the lower support, as in Fig. 92.
Let M =bending moment at mid-point of any division; bending moment at left support; T'L=vertical component of thrust at left support; component of thrust at left support; x and y=coordinates of mid-points of divisions from center of left support; ?n=moment at any midpoint of division of all exterior loads between the division and the left support.
Then, using the method of Section 167, we have EM= 0, 2:11Ix = 0, 7217y=0, and Having found the values of and the moment at any section may be calculated by Formula 25, and the line of pressure may be drawn, beginning at the left support.
The line of thrust due to change of temperature will be parallel to a line joining the ends of the arch axis.
If L =the horizontal span of the arch axis, and J=the height of its right end above its left end, using the notation of Section 16S, we have 174. Arches with Elastic Piers.—In the ordinary theory of the elastic arch, the supports are supposed to be rigid and unyielding. This can never he strictly true, but it is practically correct where good foundations are obtained and a sufficient Nveight of abutment is used.
In the construction of a series of arches, light piers are sometimes employed to carry the vertical loads, the caches being depended upon to carry the horizontal reactions. In such systems, the tops of the piers are subject to lateral motion which may materially affect the stresses in the arch rings.
The of the piers must always be designed so that the result ant of the loads fall within their middle thirds, so that the bases will remain in contact with the foundations throughout. When this is the case, the piers become cantilevers held firmly at their bases and fixed between the arches at their upper ends.
When the structure is composed of nearly equal spans, and the thrust against the pier does not differ greatly on the two sides under dead load, the effect of the flexibility of the pier may be investigated for moving loads by a method of approximation. In Fig. 93, TL and T1 are the thrusts of the spans upon the left and right respectively and R is their resultant acting upon the pier.
The maximum load is supposed upon the left span and dead load only upon the right. The difference of the horizontal components of and TR, IIR, is the horizontal com ponent of P.
This is applied at the top of the pier, causing the pier to act as a cantilever fixed at the bot tom. If we assume that the top of the pier is firmly held by the ends of the arch, so that no rotation takes place, the top of the pier will have only a horizontal motion. The effect of this motion is to lengthen the span of the arch upon the left of the pier and decrease that of the arch upon the right, which will decrease the value of and increase that of Let Q= the horizontal motion at top of pier; h=the height of pier; /,=average moment of inertia of horizontal sections of pier. The crown thrust for the span on the left then becomes, The formulas for I', and I, are unchanged by the motion of the top of the pier, and are the same as for the arch with fixed ends.
If values of and HR be found by the formula for fixed supports, and the value of Q corresponding to their difference computed, the actual value of Q will be less than the computed value, and a trial value may be used in obtaining new values of and HR, until the values of the three quantities are in fair agreement.