By combining equations (21) and (22) to eliminate MBA, and letting

Similarly, by combining Formulas (21) and (22) to eliminate MAB,

It is seen that Formulas (23) and (24) are identical with Formulas (19) and (20), except that each contains an additional term in its right-hand member. This additional term is independent of the slopes and deflections of the member, and depends solely upon the intermediate loads. Further significance is given to this term if the slopes and deflections are made equal to zero, as is true in a fixed beam with supports on the same level. The last term then becomes the resisting moment acting at the end of the fixed beam. Hence it is seen that in general the resisting moment at the end of a member with any system of intermediate loads can be expressed as the algebraic sum of two moments, one the resisting moment at the end of a member with no intermediate loads, given by Formulas (19) and (20), and the other the resisting moment at the end of a fixed beam of equal span and carrying the same system of intermediate loads.

If the resisting moment at the end of a fixed beam with supports on same level is expressed by C, with subscripts similar to those used for moments expressed by M, Formulas (23) and (24) may be written in the following general forms:

These are the general slope-deflection Formulas which apply to any tion of loading and restraint.

condi

The sign of the constant C may be determined as follows. In a fixed beam the sign of the resisting moment at the end of a member is opposite to that of the moment of external loads. For example, in Fig. 16 the moment of the resultant of the external loads about the end A is clockwise, or positive, so the resisting moment CAB is counter-clockwise, or negative; and since the moment of the resultant of the loads is counter-clockwise, or negative about B, CBA is clockwise, or positive. If the loads were upward instead of downward, the signs of CAB and CBA would be reversed. With the signs thus interpreted, C is a numerical quantity.

АВ

It has been noted that Formulas (25) and (26) apply to any condition of restraint at the ends of a member. Fig. 17 shows a member restrained at A

and hinged to the support at B, so that the resisting moment at B is zero. Formulas (25) and (26) may be written

Fig. 17.

MAB

Beam Fixed at One End and Hinged to the Support at the Other End. Intermediate Loading

If the beam is fixed at A and hinged at B, with the supports on the same level,

OA and R in this equation are zero, and the term

- (c

CBA

CAB +

represents

2

the resisting moment at the end A, and can be readily computed for any given

loading.

By similar reasoning, when the beam is restrained at the end B and hinged to the support at A,

(CAB
CAB + CBA)

be denoted

For more convenient reference, let the quantity (CAB +

САВ 2

by HAB, and the quantity (CBA + CAR) by HBA

Formulas (27) and (28) then take the general form

2

The term H represents the resisting moment at the fixed end of a beam which is fixed at one end and hinged to the support at the other, with the supports at the same level. The sign of H is determined in the same way as the sign of C in Formulas (25) and (26). That is, the sign of H is always opposite to the sign of the moment of the resultant of the external loads about the fixed end of the member. If the external loads act upward instead of downward, the signs of H in Formulas (29) and (30) are reversed.

Formulas (25) and (26) are the general equations for the ends of a member in flexure. Formulas (29) and (30) are special forms of Formulas (25) and (26), and are applicable to members having one end hinged.

It should be remembered that the signs of the quantities used in these formulas are determined by the following rules:

is positive (+) when the tangent to the elastic curve is turned in a clockwise direction.

R is positive (+) when the member is deflected in a clockwise direction. The resisting moment of the internal stresses in a section, is positive (+) when the internal mechanical couple acts in a clockwise direction with reference to the cross-section considered.

MAB

A

Fig. 18.

MBA

B

Beam Supporting any System of
Vertical Loading

If the bending moment caused by the moments of the external forces on the member, at the end at which this moment is to be determined, is positive (+), the sign before the constant in the equations is minus (-); and if this bending moment is negative (-), the sign before the constant is plus (+). For the moment at A, if the external vertical forces act downward, as shown in Fig. 18, CAB, and HAB are negative, but for the moment at B, CBA and HBA are positive.

Fig. 19. Restrained Beam with a Single Concentrated and at a distance b from the

Pab

and the distance ≈ of the centroid of the area F from B is (1⁄2) (l + b).

and

Substituting these values in the last terms of Formulas (23) and (24),

If the member is hinged instead of restrained at B, the value of HAB can be determined from the last term of Formula (27), in which

Similarly, if the member is hinged at A and restrained at B, the value of HBA can be determined from the last term of Formula (28), in which

Another condition of frequent occurrence is a loading of concentrated loads

Fig. 20.

Restrained Beam with Two Concentrated

Loads Symmetrically Placed about the Middle of Substituting this value in the

last terms of Formulas (23)

and

the Span

8

Similarly, for a member with the end A hinged, the last term of Formula (27) becomes

and for a member with the end B hinged, the last term of Formula (28) becomes

F

A geometrical meaning is attached to the term

since it represents to scale

the average length of the ordinates of the moment-diagram for a simple beam under the given loading.

From these illustrations it is seen that values of C and H may be found by Formulas (23), (24), (27), and (28). Values are also given for the more common cases of loading usually explained in text-books on strength of materials, but when so determined, the signs must be determined in accordance with the rules given on page 125.

Another method may be used to determine C and H for any kind of loading. For a member with a single concentrated load P, as shown in Fig. 19, the value Pab2

of CAB is

and the value of CBA is

12

Pa2b
12

as in Formulas (31) and (32).

If there are several concentrated loads on the member, by summation,

If there is a distributed load on the member the same general method may be used, by performing the required integration between the proper limits. Let w be the unit loading on an element of length dx, which is at a distance x from the left end, and a distance l x from the right end of the member. In

P is replaced by wdx, a by x, and b by (l — x). Then

The limits of the definite integral are fixed by the length of the loaded member.

If the unit load w is not constant, its variation may be expressed in terms of x, and the general value for the total load on a length dx thus found substituted for P in the given expression for a single concentrated load, after which the integration may be performed as indicated.

Values of C and H for different systems of loads are given in Tables I and II.