SECTION II.-Passive Strength.

1. When a weight is supported by a bar resting on two fulcrums, the pressure on each is inversely as its distance from the weight.

2. The strain on a given point of a bar, placed horizontally, and supported at both ends, from a weight placed on it, is proportional to the rectangle of the segments into which the point divides the bar.

3. Hence that strain is greatest in the middle of the bar or beam; or, in other words, if the bar be prismatic, it is most likely to break in the middle, or it is weakest there.

4. The strain produced by the weight of an equable bar, at any point of its length, is equal to the strain produced by half the weight of one segment acting at the end of a lever equal to the other segment.

5. DEF. A substance perfectly elastic is initially extended and compressed in equal degrees by equal forces, and proportionally by proportional forces.

6. DEF. The modulus of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length.

The modulus of elasticity is the measure of the elastic force of any substance.

ε

A practical notion of the modulus of elasticity may be readily obtained. Let be the quantity of a bar of wood, iron or other substance, an inch square and a foot in length would be extended or diminished by the force f; and let be any other length of a bar of equal base and like substance; then 1:::: A, or l ɛ

length 7.

ε

= A, the extension or diminution in the

The modulus of elasticity is found by this analogy as the diminution of the length of any substance is to its length, so is the force that produced that diminution to the modulus of elasticity. Or, denoting the weight of the modulus in lbs. for a base of an inch square by m; it is

ef :: 1 :m ='

And if w be the weight of a bar of the substance one inch square and 1 foot in length; then, if м be the height of the modulus of elasticity in feet, we have

7. When a force is applied to an elastic column of a rectangular prismatic form in a direction parallel to the axis, the parts nearest to the line of direction of the force exert a resistance in an opposite direction; those particles which are at a distance beyond the axis equal to a third proportional to the depth, and twelve times the distance of the line of direction of the force, remain in their natural state; and the parts beyond them act in the direction of the force.

8. The weight of the modulus of the elasticity of a column being m, a weight bending it in any manner f, the distance of the line of its application from any point of the axis D, and the depth of the column, d, the radius of curvature will be

d2 m 12 D f

9. The distance of the point of greatest curvature of a prismatic beam, from the line of direction of the force, is twice the versed sine of that arc of the circle of greatest curvature, of which the extremity is parallel to that of the beam.

When the force is longitudinal, and the curvature inconsiderable, the form coincides with the harmonic curve, the curvature being proportional to the distance from the axis; and the distance of the point of indifference from the axis becomes the secant of an arc proportional to the distance from the middle of the column.

10. If a beam is naturally of the form which a prismatic beam. would acquire, if it were slightly bent by a longitudinal force, calling its depth, d, its length, 7, the circumference of a circle of which the diameter is unity, c, the weight of the modulus of elasticity, m, the natural deviation from the rectilinear form, A', and a force applied at the extremity of the axis, f, the total deviation from the rectilinear form will be

d2 c2 ▲ m

A' = d2 c3 m — 12 l2 ƒ'
f

SCHOLIUM. It appears from this formula, that when the other quantities remain unaltered, A' varies in proportion to A, and if A = o, the beam cannot be retained in a state of inflection, while the denominator of the fraction remains a finite quantity; but when d2 cm = 12 l f, A' becomes infinite, whatever may be the magnitude of A, and the force will overpower the beam, or will at least cause it to bend so much as to derange the operation of the forces concerned. In this case f=

.8225 m, which is the force capable of holding

d2 12

the beam in equilibrium in any inconsiderable degree of cur

vature.

Hence the modulus being known for any substance, we may determine at once the weight which a given bar nearly straight is capable of supporting. For instance, in firwood, supposing its height 10,000,000 feet, a bar an inch square and ten feet long may begin to bend with the weight of a bar of the same

thickness, equal in length to .8225 ×

1

120 X 120

X 10,000,000

feet, or 571 feet; that is, with a weight of about 120 lbs. ; neglecting the effect of the weight of the bar itself. In the same manner the strength of a bar of any other substance may be determined, either from direct experiments on its flexure, or from the sounds that it produces. Iff=

✓ (.8225 n)

=

m 13 n' d3

.8225 n, and

=

.907n; whence, if we know the force required to crush a bar or column, we may calculate what must be the proportion of its length to its depth, in order that it may begin to bend rather than be crushed.

11. When a longitudinal force is applied to the extremities of a straight prismatic beam, at the distance D from the axis, the deflection of the middle of the beam will be D. (sec. arc

12. The form of an elastic bar, fixed at one end, and bearing a weight at the extremity, becomes ultimately a cubic parabola, and the depression is of the versed sine of an equal arc, in the smaller circle of curvature.

13. The weight of the modulus of the elasticity of a bar is to a weight acting at its extremity only, as four times the cube of the length to the product of the square of the depth and the depression.

14. If an equable bar be fixed horizontally at one end, and bent by its own weight, the depression at the extremity will be half the versed sine of an equal arc in the circle of curvature at the fixed point.

15. The height of the modulus of the elasticity of a bar, fixed at one end, and depressed by its own weight, is half as much more as the fourth power of the length divided by the product of the square of the depth and the depression.

16. The depression of the middle of a bar supported at both ends, produced by its own weight, is five-sixths of the versed sine of half the equal arc in the circle of least cur

vature.

17. The height of the modulus of the elasticity of a bar, supported at both ends, is of the fourth power of the length,

divided by the product of the depression and the square of the depth.

From an experiment made by Mr. Leslie on a bar in these circumstances, the height of the modulus of the elasticity of deal appears to be about 9,328,000 feet. Chladni's observations on the sounds of fir wood afford very nearly the same result.

18. The weight under which a vertical bar not fixed at the end may begin to bend, is to any weight laid on the middle of the same bar, when supported at the extremities in a horizontal position, nearly in the ratio of 15 of the length to the depression.

19. DEF. The stiffness of bodies is measured by their resistance at an equal linear deviation from their natural position.

20. The stiffness of a beam is directly as its breadth, and as the cube of its depth, and inversely as the cube of its length.. 21. The direct cohesive or repulsive strength of a body is in the joint ratio of its primitive elasticity, of its toughness, and the magnitude of its section.

Though most natural substances appear in their intimate constitution to be perfectly elastic, yet it often happens that their toughness with respect to extension and compression differs very materially. In general, bodies are said to have less toughness in resisting extension than compression.

22. The transverse strength of a beam is directly as the breadth and as the square of the depth, and inversely as the length.

SCHOLIUM. If one of the surfaces of a beam were incompressible, and the cohesive force of all its strata collected in the other, its strength would be six times as great as in the d2 m natural state; for the radius of curvature would be

Df

which could not be less than twice as great as in the natural state, because the strata would be twice as much extended, with the same curvature, as when the neutral point is in the axis; and fwould then be six times as great.

23. DEF. The resilience of a beam may be considered as proportional to the height from which a given body must fall to break it.

24. The resilience of prismatic beams is simply as their bulk.

25. The stiffest beam that can be cut out of a given cylinder is that of which the depth is to the breadth as the square root of 3 to 1, and the strongest as the square root of 2 to 1; but the

most resilient will be that which has its depth and breadth equal.

26. Supposing a tube of evanescent thickness to be expanded into a similar tube of greater diameter, but of equal length, the quantity of matter remaining the same, the strength will be increased in the ratio of the diameter, and the stiffness in the ratio of the square of the diameter, but the resilience will remain unaltered.

27. The stiffness of a cylinder is to that of its circumscribing prism as three times the bulk of the cylinder to four times that of the prism.

28. If a column, subjected to a longitudinal force, be cut out of a plank or slab of equable depth, in order that the extension and compression of the surfaces may be initially everywhere equal, its outline must be a circular arc.

29. If a column be cut out of a plank of equable breadth, and the outline limiting its depth be composed of two triangles, joined at their bases, the tension of the surfaces produced by a longitudinal force will be everywhere equal, when the radius of curvature at the middle becomes equal to half the length of the column; and in this case the curve will be a cycloid.

When the curvature at the middle differs from that of the cycloid, the figure of the column becomes of more difficult investigation. It may, however, be delineated mechanically, making both the depth of the column and its radius of curvature proportional always to a. If the breadth of the column vary in the same proportion as the depth, they must both be everywhere as the cube root of a, the ordinate.-(Young's Nat. Phil. vol. ii.)

30. The modulus of elasticity has not yet been ascertained in reference to so many subjects as could be wished. Professor Leslie exhibits several, however, as below. That of white marble is 2,150,000 feet, or a weight of 2,520,000 pounds avoirdupois on the square inch; while that of Portland stone is only 1,570,000 feet, corresponding on the square inch to the weight of 1,530,000 pounds.

White marble and Portland stone are found to have, for every square inch of section, a cohesive power of 1811 lb. and 857 lb.; wherefore, suspended columns of these stones, of the altitude of 1542 and 945 feet, or only the 1394th and 1789th part of their respective measure of elasticity, would be torn asunder by their own weight.

31. Of the principal kinds of timber employed in building and carpentry, the annexed table will exhibit their respective Modulus of Elasticity, and the portion of it which limits their cohesion, or which lengthwise would tear them asunder.