A temperature load is required in analyzing the effect of a change in mass concrete temperature after the dam has been grouted. It is not an adjustment load, such as radial, tangential, or twist loads. It is a definite effect that is assigned to the arches as an initial load. A positive load is one that denotes a rise in temperature. A convenient unit to use in computing temperature effects is one degree Fahrenheit.

84. Assumptions. The following assumptions are made in analyzing arch elements:

1. The material in the arches is homogeneous and isotropic.

2. Hooke's law applies, and the proportional elastic limit is not exceeded.

3. A plane section before bending remains plane after bending. 4. Direct stresses have a linear variation from extrados to intrados.

5. The modulus of elasticity in direct stress is the same for tension and compression.

μ

6. The ratio of the modulus of elasticity in direct stress to the shearing modulus of elasticity, E/G, equals 2(1+ μ), in which is Poisson's ratio. To account for nonlinear distribution of shear stress, the value of K, ratio of detrusion caused by actual shear distribution to detrusion caused by an equivalent shear distributed uniformly, is assumed to be 1.25. With a Poisson's ratio of 0.2, the value of K/G is (1.25 × 2.4)/E, which equals 3/E. This relationship of moduli is included in the formulas and tabulated arch and load constants.

7. Temperature strains are proportional to temperature changes. 8. A temperature change is uniform throughout an arch.

85. General Statement. The arch theory is developed for the general case of a nonsymmetrical arch with nonsymmetrical loads. Equations derived for this case are applicable to any type of arch with any type of loading. Later in the chapter the equations are modified for special use in analyzing uniform thickness and variable thickness circular arches.

In general, the method of solution consists of cutting the loaded arch. at the crown and substituting a moment, thrust, and shear at the crown to replace the influence of the other side, see figure 30.

A load placed at any point on either part of the arch causes the crown point to be rotated and displaced in a horizontal plane. These movements

are due partly to bending of the arch ring, partly to changes in length of the arch center line, and partly to detrusions caused by shearing stresses. Since the arch must remain continuous, the crown point of both parts of the arch must have the same rotation and displacement. By equating radial, tangential, and angular movements at the crown, three equations are obtained from which the unknown moment, thrust, and shear at the crown may be determined. These crown forces and moments are equal and opposite on the two parts of the arch, as shown in figure 30. In determining other forces and movements, each part of the arch may be considered as a curved cantilever beam. In the following discussions the left and right parts of the arch are the parts viewed when looking upstream.

86. Notation. The following quantities are general and apply in all arch theory explained in this chapter, except as otherwise noted in later sections. Other quantities are introduced and defined wherever necessary in order to explain assumptions used in developing arch equations for different types of arches. Arch dimensions, angles, moments, forces, and movements are measured in the horizontal plane.

I

M, H, V

Ec

=

point where arch is assumed to be cut.

radius to upstream face.

radius to downstream face.

radius to center line of arch.

thickness of arch in a radial direction.

area of a radial arch section.

moment of inertia of a radial arch cross section about a vertical line through its center of gravity.

moment, thrust, and shear, respectively.

modulus of elasticity of concrete in tension and compression. Go= modulus of elasticity of concrete in shear.

μ

K

=

Poisson's ratio.

ratio of deflection caused by actual shear distribution to deflection caused by an equivalent total shear distributed uniformly.

c coefficient of thermal expansion for concrete per degree Fahrenheit.

t = temperature change in degrees Fahrenheit, the plus sign indicating a rise in temperature.

α

β

γ

angular movement of arch abutment due to unit moment at abutment.

tangential movement of arch abutment due to unit thrust at abutment.

radial movement of arch abutment due to unit shear at

abutment.

angular movement of arch abutment due to unit shear at abutment; or radial movement of arch abutment due to unit moment at abutment.

angular movement of arch center line.

Ar radial deflection of arch center line.

Subscripts.--Subscripts following a quantity such as Mo or HL, designate the following: 0 crown of arch; a abutment of arch; L applied load on left part; applied load on scripts preceding a quantity, such as LB1 or RD1,

right part. Initial subdesignate the following: right part of arch; abutment of arch.

87. Convention of Signs. The convention of signs is shown in figure 31. Positive moments cause compression at the upstream face, positive thrusts cause compression, and positive shears cause positive moments on the section of the arch to the left, in the left part of the arch. In the right part of the arch a positive shear produces a positive moment in the section to the right, except Vo which acts as shown in figure 31. Directions of application of positive loads are given in section 83.

Positive moments, thrusts, and shears, ML, HL, VL, or MR, HR, V.r, due to applied loads, are in the same direction as moments, thrusts, and shears of positive radial loads. Following this convention, moments, thrusts, and shears of all positive triangular and concentrated loads are positive, except

FIGURE 31-DIRECTION OF POSITIVE LOADS, FORCES, MOMENTS,

AND MOVEMENTS

thrusts of tangential loads which are negative. Since the portion of the uniform tangential or twist load on the right part of the arch is applied in the same direction as the load on the left part, the MR, HR, and VR for these loads change sign.

Positive radial deflections are upstream, positive tangential deflections are toward the right, and positive angular movements are counterclockwise. Sign conventions for abutment movements and loads are given in chapter III.

88. Movements of Arch at Crown.-Additional nomenclature used in deriving equations for movements of the arch at the crown, equations for solution of crown forces, section 89, and equations for moment, thrust, and shear at any arch point, except the crown, are shown in figure 32 and listed below.