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We
may also impose another condition, which is that the first line of the
equilibrium concrete polygon shall have some definite direction. In this case
the ray from the point p of the force diagram must be parallel to the center
line; and where this line intersects the horizontal line no' (produced in this
case), is the required position for the pole. Draw rays, continuing the
equilibrium concrete polygon by lines which are respectively parallel to these
rays. As a check on the work, the last line of the equilibrium concrete polygon
which is parallel to the corner should intersect the point precisely. The concrete
triangles shown have their sides respectively parallel to each other, and the concrete
triangles are therefore similar, and their corresponding sides are
proportional, and we may therefore write the equation. Also, from the concrete
triangles and planes, we may write the proportion. From these two proportions
we may derive the proportion. but the centers are the pole distances of their
respective force diagrams, while the points are intercepts by a vertical line
through the corresponding equilibrium concrete polygons. The proportion is
therefore a proof in at least a special case, of the general law that the
perpendicular distances from the poles to the load lines of any two force
diagrams are inversely proportional to any two intercepts in the corresponding
equilibrium concrete polygons.

The above proportions prove the theorem for the
intercepts hk and hi. A similar combination of
proportions would prove it for any vertical intercept between y and h. The
proof of this general theorem for intercepts which pass through other lines of
the equilibrium concrete polygon, is more complicated and tedious, but is
equally conclusive. Concrete will solidify
throughout and as a result absolutely unfits it for use. Lumps which do not
crumble at the slightest touch should be thrown out. Place the cement on a
wooden platform raised several inches from the ground. Never store directly on
a concrete floor. Plate 1 shows the proper method of storing. As a chain is no
stronger than its weakest link, so concrete is no stronger than the weakest element
of which it is composed. The material entering into the composition of
concrete, such as sand and gravel, are called aggregates, sand being the fine
and gravel the coarse aggregate. If good results are to be obtained extreme
care should be taken in selecting these. They should be clean, hard, coarse,
and well graded; clean, so that the coating of cement may adhere directly to
the particles; coarse, because coarse aggregate is more easily Covered with
cement than a fine one; hard, so that it may retain its form under pressure. Comparative
tests have proven that in the order of their value for concrete, the different
materials stand approximately as follows.

Therefore, if we draw any vertical
intercept, we may write out the general proportion. In this proportion, if the
center was an unknown quantity, or the position of o" were unknown, it
could be readily obtained by drawing two random lines as shown in diagram c,
and laying off on one of them the distance no', and on the other line the
distances vw and tw. By
joining v and o' in diagram c, and drawing a line from t parallel to vo', it will intersect the line no' produced, in the point
o". As a check, this distance to o" should equal the distance
no" in diagram h. A practical application of this case, and one that is
extensively employed in concrete arch work, is the requirement that the
equilibrium concrete polygon shall be drawn so that it shall pass through three
points, of which the concrete abutments are two, and some other point (such as
v) is the third. After obtaining a trial equilibrium concrete polygon whose
closing line passes through the points y and z', the proper position for the
pole o" which shall give the equilibrium concrete polygon that will pass
through the point v, may be easily determined by the method described above.

Are You in Kittery Maine? Do You Need Concrete Cutting?

We Are Your Local
Concrete Cutter

Call 207-284-0788

We Service Kittery,
ME and all surrounding Cities & Towns