The bending moment at the end of the gage length (X =0) is as follows:

x =

The bending moment at the center of the gage length (X=

The moment diagram between the third points when there is both positive and negative bending moment in the gage length is shown in fig. 3, in which xx' is the horizontal axis of the moment diagram. The curve d, e, e', d' is a parabola, and crosses the axis at two points, viz, at e and at e', between the ends of the deformeters. Then in the gage length c c' there is negative bending moment from c to e and from e' to c', and positive bending moment from e to e'. The dotted lines c d, c' d', and d d' are drawn for the purpose of demonstration. Then the distance M. represents the bending moment at the center of the gage length, and M. represents the bending moment at the end of the gage length. The negative bending-moment areas within the gage length are c d e and c' d' e', each being represented by B. The positive bending moment area within the gage length is e Fe', and is represented by A.

The condition that the positive bending-moment area is equal to the negative bending-moment area is represented by the equation: -2B.

A

==

Adding the quantity -C to both sides of the equation gives: A+ (C) -2B-C.

=

The first part of this equation is the area included between the horizontal line d d', and the parabola d F d', that is:

The second part of the equation is equal to the area of the rectangle dc c' d', that is:

Substituting the values of M, and M. as found above gives—

In almost all the beams tested at the laboratories L, Z, g, and m are constant.

It only remains to find W and to compute R. A table has been computed by the above formula for all the usual values of W, and the corresponding values of R in any case can be read directly from the table.

Form K (p. 55) is used for reporting the results of the tests on plain concrete beams.

The breaking load consists of the applied load together with the weight of the beam plus the deformeters. The total load is read directly from the scale arm of the machine when the beam fails, while the applied load is the total load less the weight of the beam and deformeters, which are found at the beginning of the test.

After the loads have been found the bending moment is computed and the value of the arbitrary term bd is found for purposes of

M

M plotting. The term bd2 at the center for the breaking load is obtained by the usual calculation, considering the weight of the beam between the supports and the 6-inch overhang at each end as a uniform load and considering the weight of the deformeters as two loads. concentrated at the ends of the gage length.

The quantities recorded for unit elongation of the lower fiber for weight of beam, together with the weight of all attachments, are the micrometer readings (1) when the beam is partly suspended so as to cause zero total deformations in the gage length, and (2) when the beam rests under its own load plus that of the deformeters.

The computations involved are the subtraction of these two micrometer readings, the averaging of the differences obtained on opposite sides of the beam, and the correction of this average so that it will represent the elongation of the lower fiber instead of that of the fiber upon which the deformeter was clamped. This last computation is made upon the basis that the elongation varies uniformly

as the distance from the neutral axis or upon the usual assumption of the conservation of plane sections. The reading of the two sets of deformeters verifies this assumption.

The unit deformation is then obtained by using the parabolic formula, except in cases where the bending moment due to the applied load is so great in comparison to that due to the weight of the beam that the error due to dividing by the gage length is less than the probable error in reading the deformeters. The correction by use of the parabolic formula is based upon the assumption that the total deformation of any fiber is proportional to the product of the bending moment and the length of the fiber, or, in other words, to the bending-moment area included in the length of the fiber. This is represented in fig. 4, in which Mg is the bending moment at the end

FIG. 4.-Diagram illustrating bending moment between gage points.

2

of the gage length and M. is that at the center of the gage length, the difference being M. The area (A) in the diagram is equal to A=g (Mg +Mw). Dividing this area by the greatest M. gives a new gage length which, were the bending moment constant over it and equal to Me would give the same total deformation which was measured. Dividing this deformation by the new gage length gives the unit deformation where the bending moment is greatest. The final deformeter values are calculated from the load and the micrometer readings at the last full set of micrometer readings before the maximum load was reached.

The percentage of the distance of the neutral axis from the top of the beam is assumed to be equal to the deformation of the top fiber multiplied by 100 and divided by the sum of the top and bottom deformations.

The modulus of rupture is calculated by means of the formula Me, the value of M at the center of the span being used.

S=

I'

The short sections of the plain beams are not suspended for zero deformations in the gage length, and therefore the deformations calculated for these are those due to the applied load only.

Form L (p. 57) is used for reporting the results of the tests of reinforced concrete beams.

The percentages of steel recorded in the batch report are given in terms of the section of concrete above a line drawn through the centers of the rods, the lower layer being taken when there is more than one layer.

The position of the neutral axis is calculated as in the plain beams, except that instead of using the deformation of the lower fiber, the deformation of the steel is used, thus obtaining the percentage of the depth below the top in terms of the distance from the top of the beam to the center of the lower layer of rods. The position of the neutral axis is calculated for several loads up to the maximum, and curves are drawn in order to show the variation in the position with the increase in the load.

The values under this general heading are obtained from the load, deformations, and deflections at the last full set of micrometer readings before the maximum load. After the location of the neutral axis has been found, the final deformeter values at the top of the beam are corrected so as to give the deformations at the extreme top. It should be noted that the lower micrometers are clamped directly over the steel, and therefore no correction of the micrometer readings is necessary to allow for the fact that the fiber whose elongation is required is not the fiber upon which the micrometers are clamped.

All the calculations made for the plain beams are repeated here except the one giving the modulus of rupture, for which a special formula must be used. The maximum values are obtained from the load, lower micrometer readings, and deflections when the beam has reached its maximum resistance, or from the last full set of deformeter readings before failure.

TESTS OF CYLINDERS AND CUBES.

Method.-A cylinder and a cube are made from the same batch of concrete from which each beam is molded, and all are tamped by hand with a tamper weighing 7 pounds and having a circular face 3 inches in diameter. The same molds (shown in Pl. VIII, B, p. 30) are used for these specimens as for those tested in the constituent-materials section. The method of testing is also the same (pp. 31-36). The compressometers for measuring the deformations. of the cylinders are shown in Pl. IX, C' (p. 32). The micrometers used on these compressometers measure directly to 10 inch. Form M is used for recording the results of tests.

Form M. {

Cylinder reg. No.

......

Length,

inches;

mate strength,

value,

machine,

at

UNITED STATES GEOLOGICAL SURVEY.
STRUCTURAL-MATERIALS TESTING LABORATORY.

Lab. No.
Weight,

inches.
cubic feet. Weight,

CYLINDER

TEST.

inches.

cubic

pounds. Ulti

Gage length, ...... inches. Diameter,
pounds. Area,
square inches. Volume,
pounds per cubic foot. Ultimate load,
pounds per square inch. Initial coefficient of elasticity,
pounds per square inch. Probable ultimate unit deformation,

.......

......

Range of linear
Bedding in

.......

time

For character of concrete and corresponding test piece see Batch report Bm Cylinders and cubes brought from damp closet at......; weighing, measuring, and capping finished time......; test of cylinders started at ......; test of cylinders completed at .: sheet handed to office at Remarks.-.

Computations. The compressive strengths of the cubes and the cylinders are calculated in pounds per square inch. For the cylinders the modulus of elasticity is determined by drawing a curve showing the values at different loads. A tangent to the curve at or near its origin is assumed to represent the initial modulus of elasticity.

BOND TEST PIECES.

Method. The schedule of bond tests is shown in the table on pages 40-47. A bond test piece in the machine ready for testing is shown in Pl. XVIII, B. The concrete cylinder is placed on top of the machine with the embedded rod projecting downward. The lower end of the rod is gripped in the jaws of the machine. The lower surface of the concrete cylinder is embedded in plaster of Paris on a 1-inch plate with a hole in its center one-sixteenth inch greater in diameter than the rod which passes through it. The instrument for measuring the slip of the rod is shown at the top of the test piece in the figure. During the test the micrometer and load are read at intervals of about 500 pounds until the slip of the rod amounts to about one-tenth inch. The load in all cases is applied continuously until failure. The bond pieces are tested at the ages of 30, 90, 180, and 360 days.

Computations.-Form N is used for recording results of the bond

The unit bond at any load is found by dividing the load by the surface area of the rod in contact with the concrete.