Suppose that R, is less than L, of A, i. e., that the external shear for the section is negative; then, if vertical pulls be applied at the cut ends, upward on A and downward on B, the beam will stand under its load and in a horizontal position, just as a solid beam. These pulls can be supplied, in the model of the beam, by means of a cord S tied to two brackets fastened on A and B, as shown. In the solid beam the two parts act upon each other directly, and the vertical forces are shearing stresses, since they act in the plane of the surfaces to which they are applied.
59. Relation Between the Stress at a Section and the and Reactions on Either Side of It. Let Fig. 35 represent the portion of -a beam on the left of a section; and let R, denote the left reaction; L, and L, the loads; W the weight of the left part; C, T, and S the compression, tension, and shear respectively which the right part exerts upon the left.
Since the part of the beam here represented is at rest, all the forces exerted upon it are balanced; and when a number of hori zontal and vertical forces are balanced, then 1. The algebraic sum of the horizontal forces equals zero.
2. " " " " " vertical " " " 3. " " " " " moments of all the forces with respect to any point equals zero.
To satisfy condition 1, since the tension and compression are the only horizontal forces, the tension must equal the compression,. To satisfy condition 2, S (the internal shear) must equal the algebraic sum of all the other vertical forces on the portion, that is, must equal the external shear for the section; also, S must act up or down according as the external shear is negative or positive. In other words, briefly expressed, the internal and external shears at a sect-ion are equal and opposite.
To satisfy condition 3, the algebraic sum of the moments of the fibre stresses about the neutral axis must be equal to the sum of the moments of all the other forces acting on the portion about the same line, and the signs of those sums must be opposite. (The moment of the shear about the neutral axis is zero.) Now, the sum of the moments of the loads and reactions is called the bend ing moment at the section, and if we use the term resisting mo ment to signify the sum of the moments of the fibre stresses (ten sions and compressions ) about the neutral axis, then we may say briefly that the resisting and the bending moments at a section are the two moments are opposite in sign.
6o. The Fibre Stress. As before stated, the fibre stress is not a uniform one, that is, it is not uniformly distributed over the section on which it acts. At any section, the compression is most
" intense" (or the unit-compressive stress is greatest) on the con cave side; the tension is most intense (or the unit-tensile stress is greatest) on the convex side; and the unit-compressive and unit tensile stresses decrease toward the neutral axis, at which place the unit-fibre stress is zero.
If the fibre stresses are within the elastic limit, then the two straight lines on the side of a beam referred to in Art. 57 will still be straight after the beam is bent; hence the elongations and short enings of the fibres vary directly as their distance from the neutral axis. Since the stresses (if within the elastic limit) and deforma tions in a given material are proportional, the unit-fibre stress varies as the distance from the neutral axis.
Let Fig. 36a represent a portion of a bent•beam, 36b its cross section, nn the neutral line, and NN the neutral axis. The way in which the unit-fibre stress on the section varies can be rep resented graphically as follows: Lay off ac, by some scale, to represent the unit-fibre stress in the top fibre, and join c and n, extending the line to the lower side of the beam; also make be' equal to be" and draw yc' . Then the arrows represent the unit-fibre stresses, for their lengths vary as their distances from the neutral axis.
6i. Value of the Resisting Moment. denotes the unit fibre stress in the fibre farthest from the neutral axis (the greatest unit-fibre stress on the cross-section), and c the distance from the neutral axis to the remotest fibre, while S„ S„ S„ etc., denote the unit-fibre stresses at points whose distances from the neutral axis are, respectively, y„ y„ etc. (see Fig. 366), then S : S, : or S = TY,• S Also, o, = S T etc.
Let a„ a„ a„ etc., be the areas of the cross-sections of the fibres whose distances from the neutral axis are, respectively, y„ y„ y„ etc. Then the stresses on those fibres are, respectively, S, a„ S2 a„ S, a„ etc.; S S S Or, 7 ---3f2a21 etc.
The arms of these forces or stresses with respect to the neutral axis are, respectively, y„ y„ etc.; hence their moments are S 'S T aM, etc., and the sum of the moments (that is, the resisting moment) is y,2 c + etc. =— c (a, A -I- a, yl + etc.) Now a, y; etc. is the sum of the products obtained by multiplying each infinitesimal part of the area of the cross-section by the square of its distance from the neutral axis; hence, it is the moment of inertia of the cross section with respect to the neutral axis. If this moment is denoted by I, then the value of the resist ing moment is SI