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IN MEMORIAM 
 FLOR1AN CAJOR1 
 
THIIE 
 
 MECHANICAL ERRORS 
 
 -IN- 
 
 Tfie Common Ttieonj of Flexure 
 
 5 
 c- 
 
THE 
 
 MECHANICAL ERRORS 
 
 -IN- 
 
 Tfie Common THBoru of Flexure 
 
 BY- 
 
 E. H. [COUSINS, 
 
 C1VII, ENGINEER. 
 
 AUSTIN, TEXAS: 
 
 EUGENE VON BOECKMANN, PRINTER AND BOOKBINDER. 
 ' 1891. 
 
CAJORI 
 
General Principles Applicable to 
 Balanced Parallel Forces. 
 
 Before discussing the mechanics involved in the "com- 
 mon theory of flexure," it is necessary to restate some ele- 
 mentary principles relating to the action of forces, and to 
 define certain distinctions in technical terms, drawn from 
 practical mechanics, that are not found in text-books of the 
 subject. 
 
 i st. In a balanced system of parallel forces the sums of 
 the forces acting in opposite directions must be zero ; in 
 other words, the sum of the vertical forces must be zero and 
 the sum of the horizontal forces must be zero. 
 
 2nd. The algebraic sum of the moments relative to any 
 axis shall be nothing. 
 
 3d. When two equal and opposite forces balance, whose 
 lines of action do not coincide, a shearing force must be 
 transmitted from the line of action of one to that of the 
 other, this shearing is the right line balancing agent. 
 
 4th. The lever-arm of a force is the distance traveled by 
 its shearing force in opder to reach a centre of motion or 
 tendency. 
 
 5th. A shearing force cannot be transmitted by a body 
 through and beyond a center of rotation or tendency. 
 
 6th. A shearing force has no leverage and therefore no 
 moment. 
 
 7th. A force to possess leverage must transmit its shear- 
 ing force to a center of rotation or tendency, where it must 
 be opposed by an equal and opposite force. 
 
 8th. A body, and therefore any part of the body, is at 
 
rest in any direction, when each particle of one of its planes 
 is solicited to move in opposite directions by pairs of equal 
 and opposed conspiring forces, parallel to the given direction. 
 
 9th. Two equal and opposed shearing forces will pro- 
 duce compression in, and if sufficiently intense, crush the 
 body that transmits them. 
 
 loth. When an unbalanced force is applied to a beam 
 and equilibrium is established by the development of inter- 
 nal forces, the originally unbalanced force must be equili- 
 brated by its own generated components, and not by those 
 of some other applied force. 
 
 The ist and 2nd conditions are those usually given as 
 solely sufficient to determine the conditions of equilibrium of 
 any body, the others are none the less universally true, and 
 are here formulated for the convenience of reference, as their 
 existence is very frequently ignored by writers, in their dis- 
 cussion of mechanical problems. 
 
 Moments. The moment of a force about a point meas- 
 ures the tendency of the force to produce rotation about the 
 point, when one point of the body is "fixed" the tendency 
 to rotation is around this point, which is therefore the origin 
 of moments. 
 
 Theoretical mechanics takes the moment of a force with 
 respect to any point, whether there is a tendency to rotate 
 around it or not, in the first case the moments possess lever- 
 age or mechanical power, in the second they do not; theo- 
 retically, the moment of a force may have an infinite num- 
 ber of values, but in practical mechanics it has value only 
 when taken with reference to a center of rotation or tendency 
 in the body to which it is applied. 
 
 Theoretical moments, from an "anywhere origin," are 
 sufficient to determine the conditions under which a whole 
 body may be at rest, relatively to the other bodies of the 
 world; but when we equate power and resistance, as in a 
 solid beam, the moments must possess mechanical power, 
 such as Archimides would have used had he turned the 
 world over, which power they only have when taken with 
 
reference to a common center \ around which they all tend to 
 rotate, and around which the body could rotate if the forces 
 that established the condition of tendency were removed. 
 The location of this center clearly classes our forces into 
 poivet and resistance, the failure to do this in the "common 
 theory" causes the tension to be the resistance, and \hepower 
 to be the load and the compression, as their moments add, 
 having like signs, which of itself condemns the theory. 
 
 In nature, the moments of two applied forces, whose 
 signs are unlike, never add, as the mechanical power ex- 
 pressed by each moment tends to do opposite or unlike 
 things. The only exception to this invariable rule is on 
 paper, where the moments of the unbalanced forces of a 
 static couple are said to add ; but the moments of these 
 forces possess no mechanical power, only a possibility of 
 such, which is only realized when the forces balance as 
 right line forces, and then one force has no moment as its 
 lever-arm is zero. The application of this mythical mechan- 
 ical moment of the unbalanced forces of a couple to the 
 tension and compression in a solid beam has lead to a palpa- 
 ble violation of the fundamental principle above stated. 
 The moment given a static couple violates the condition of 
 tendency which is the very essence of moments. 
 
 Shearing Force and Leverage. Shearing is a peculiar 
 auxiliary force that is found to exist when an applied force 
 has leverage ; it travels from the point of application of the 
 force to the center of rotation or tendency to such rotation, 
 it enables the force to possess leverage and to balance an- 
 other opposed equal force when their lines of action do not 
 coincide. 
 
 At any section between the point of application and the 
 center of rotation, it is equal in intensity, parallel to the 
 force, and has the same direction its tendency is to cross- 
 cut the body that transmits it at each sectional plane. 
 
 A shearing force cannot possess leverage, its existence 
 being the result of leverage, it cannot in turn reproduce its 
 own cause. ' 
 
The forces that have a shearing force are purely static, 
 such as a single force that tends to produce rotation, and a 
 pair of opposed tensile forces that tend to resist rotation, 
 if these forces have no shearing they neither tend to produce 
 nor to resist rotation ; for they then have no leverage. 
 But neither of the two opposed compressive forces have any 
 shearing force, and the leverage they possess is direct with- 
 out its assistance. A compressive force has this leverage 
 in resisting the rotating tendency of another force, only 
 when their tendencies are in opposite directions around an 
 actual (not theoretical) center of tendency to rotation. In 
 Fig. i, if the actual center of rotation is below the point 
 of application of the compressive force C, as it really is, 
 then the force C has no leverage in resisting the rotating 
 tendency of the load I,. 
 
 Observation teaches that a body cannot transmit a shear- 
 ing force a greater distance, than that, from the line of action 
 of the force to the center of rotation, and that this distance 
 traveled by the shearing is the limit to the length of the 
 lever-arm of the force. 
 
 The beam A, Fig. i presses the support S through the 
 shearing force I, and the support presses the beam with an 
 equal and opposite force, as this force S may also be the 
 shear of another force, these two equal and opposed shears 
 will produce pressure equal to that had the two forces 
 been" directly applied at S. A familiar illustration of this 
 is seen in the ordinary nut-cracker, the pressure on the nut 
 is the sum of the shears of the force applied by the hand 
 and the force developed at the rivet of the hinged joint. 
 
 Couples. Two equal and opposite unbalenced facts, whose 
 lines of action do not coincide constitute a static couple, 
 and the bod)'' to which it is applied should be in equili- 
 brium, as the ist condition is rigidly fufilled, but the ef- 
 fect of a couple is to rotate the body about its center of 
 mass, hence it is not in equilibrium. However, the in- 
 stant the body, from any cause, is enabled to transmit a 
 shearing force, or fulfill our 3d condition, equilibrium is es- 
 
tablished, but the pair of forces cease to form a couple and 
 become a part of a common lever where one force is the 
 power, the other the fulcrum and the resistance whatever 
 enabled the body to transmit a shearing force. 
 
 A "static couple" is a misnomer, the rotation produced 
 by it about the "center of mass" is dynamic, which cannot 
 be converted into a tendency about the same "center of 
 mass' ' for the instant it becomes static equilibrium the cen- 
 ter of tendency to rotation shifts from the center of mass to 
 the point of application of one of the forces that previously 
 formed the "static couple." The application of the "parel- 
 lelogram of forces" to the equilibrated forces will demon- 
 strate the truth of this, or it fulfills our yth condition. 
 
 Theoretical mechanics considers any two equal and oppo- 
 site forces ' to form a couple that either rotates or tends to 
 rotate the body about its center of mass, whether the forces 
 are balanced as right line forces or not, ' This is evidently 
 erronious, for if the forces are balanced it is physically im- 
 possible for the body to transmit the shearing force (the 
 right line balancing agent) through a center of rotation or 
 tendency to such rotation, it can transmit it to such a cen- 
 ter, but not through it, hence if the forces tend to turn the 
 body about its center of mass they are unbalanced. This 
 is an especially important distinction to make, since a 
 writer, who is usually very clear, states that the forces (I, 
 S) Fig. i form a couple that tends to rotate the body A, 
 about its center of mass, thus making the shearing force L 
 travel through a center of tendency to rotation, and a point 
 that is "fixed" by the opposite forces, S, and shearing force 
 I,, tend to travel in the arc af a circle around this center, 
 both of which being a mechanical impossibility. The rule 
 of statics so often quoted by authors in the analysis of the 
 common theory, that "if two couples applied to the same 
 body balance each other they must have opposite algebraic 
 signs," and its corallary, if the forces in the direction of one 
 axis reduce to a couple those in the direction of the other 
 must reduce to a couple also, for equilibrium has led them 
 
to erronious conclusions. The rule is only true when the 
 pairs of forces balance as right line forces, and thus cease 
 to be static couples, and the resultant moment of each pair 
 is the leverage moment of a single force and its corallary, 
 when the resultant moment of one pair reduces to the mo- 
 ment of a single force, the resultant moment of the other 
 pair must reduce to the leverage moment of a single force 
 with opposite sign. 
 
 To illustrate (L S) and (T C) Fig. i are said to form 
 such couples. Opposed to S there is a shearing force Iy 
 equal to it, our couple (I, S) in balancing the forces has 
 the vertical effect of three forces, the resultant moment of 
 which, according to a rule of statics, is the moment of the 
 single force L,. In the same way, horizonally, we have the 
 effect of three forces, T and C and shearing force T, equal 
 and opposed to C, the resultant moment of the three is the 
 moment of the single force T. 
 
 When the resultant moment in the direction of one axis 
 does not reduce to that of a single force, as the tension and 
 compression in a beam, the mechanical necesssity for and 
 their ability to develop equally does not exist. In this case 
 the line of direction of neither force passes throngh the cen- 
 ten of rotation. The line of action of the compressive force C 
 being between the line of action of its companion force T 
 and the center of rotation, the point of support, its moment 
 and that of the rotating force must have opposite signs, and 
 T must have the same sign and direction of C, if it is an 
 "applied" force. 
 
 THE COMMON THEORY OF FLEXURE. 
 
 It is a misnomer to call the common theory ' 'a theory of 
 flexure' ' for the theory itself does not recognize the effect 
 that is produced upon the interior of the beam by bending 
 in the slightest degree. It is true from knowledge obtain- 
 ed outside of the teaching of the theory, the total amount 
 of the interior forces is derived from considering them to 
 vary uniformly from a neutral line, but this is also done 
 
when the beam is not bent at all, then it can no more be 
 called a ' 'a theory of flexure' ' than it is of any other struc- 
 ture in which the stresses are equal, such as bridge trusses, 
 etc. 
 
 To be a theory of flexure it must recognize that the stres- 
 see before flexure begins, exist in a different condition from 
 that after, and that flexure caused them to change this con- 
 dition and to assume the order of arrangement that is 
 known to exist in a bent beam. 
 
 In its mechanics the common theory is based up undis- 
 puted principles of pure statics, but when it is attempted to 
 explain the phenominon known to attend flexure it can 
 only be done by adopting, at every point, principles just 
 the reverse of its teachings, and it is a mechanical error to 
 attribute to it effects that it cannot produce. 
 
 In reasoning from an effect to its cause the human mind 
 more readily accepts false logic as^true than when it adopts 
 the reverse process. The best authorities emphasize the 
 statement "that a body in static equilibrium is conceived to 
 be incaple of bending, breaking or changing its shape in any 
 way, and their effect is not considered," yet from the 
 knowledge of what takes place in "bending," "those 
 who ought to know how to apply principles" to effect has 
 described its cause as being ' 'a clear case of statics' ' when 
 the reverse process of reasoning would have denied the ef- 
 fect claimed at every point. To the extent that a 
 beam rotates or bends under a transverse or a column de- 
 flects under a longitudinal load, it is actuat constrained rota- 
 tion, this induces actual longitudinal molecular movement, 
 which, of itself, takes it out of the jurisdiction of static 
 laws. Static equilibrium is only a tendency to rotation, 
 which rotation, if it took place, would be free without mo- 
 lecular movement. 
 
 Static forces acting in a single plane are always equili- 
 brated on the principle of the parallelogram of forces, when 
 their resultants intersect. In the parallelogram only two 
 forces possess mechanical power, and the equality of their 
 
10 
 
 moments prevents rotation, but the common theory con- 
 ceives that three forces have this power ; as the compres- 
 sion force simply prevents horizontal translation, it may be 
 left out, just as the companion force that prevents vertical 
 translation, as neither have any lever-arm. 
 
 In the "parallelogram of forces" the center of tendency 
 to rotation is at the end of the diagonal where the forces are 
 balanced by their reactions, which in the solid beam is a 
 point on the resultant of the compressive force, but there 
 can be no rotation around this point and therefore no ten- 
 dency as the two are inseparable, which destroys the com- 
 mom theory equilibrium, as it must be around some point 
 outside of the parellelogram. 
 
 Generated forces that produce compressive strain and 
 possess mechanical power or leverage are not recognized in 
 statics, and its treaties make no attempt to trace the me- 
 chanical relation that must exist between the generating 
 and generated forces, therefore the rules that formulate 
 the principles of statics do not apply when this is the case, 
 or more than two intersecting forces possess mechanical 
 power in an equilibrated system. 
 
 The "common theory of flexure" is deduced from the 
 following conditions. L,et a beam be loaded at its ends 
 and supported at the center, then after equilibrium has 
 been established conceive one-half of the beam to be re- 
 moved, and the balanced condition destroyed. Then it is 
 conceived that we have, only, an unbalanced force L, and 
 a rigid body A, and that we must by external applied 
 forces re-establish, de novo, the equilibrium that existed be- 
 tween the external force I, and the developed internal 
 forces, which we destroyed in the cutting process. 
 
 This gives the diagram of forces represented in Fig. i. 
 From the ist and 2nd conditions given, we have L,=S, and 
 T=C, and the moment of the (L,=S), system equal to the 
 moment of the (T=C) system. 
 
 That T must equal C as the sole and sufficient cause of 
 equilibrium is only true when the forces are concurrent and 
 
11 
 
 their lines of action coincide, but here a distance intervenes 
 between them, and in order to balance they must get to- 
 gether in some manner, and be opposed just as if they 
 were concurring and their lines of action coincided, this 
 the beam A enables them to do by transmitting a longitudi- 
 nal shearing force from T to C. 
 
 To follow the line of argument of a writer 1 on this sub- 
 ject conceive that we have only the forces L and T, verti- 
 cally, the ist condition must be fulfilled, viz: "L,=S since 
 the pressure upon the fulcrum must equal the dormant 
 force L," or its shearing force. "To prevent rotation we 
 
 A 
 
 H- 
 
 - C 
 
 \/ 
 
 I 
 
 Fig. 1 
 
 must now have 1,1= Td. But these two conditions are not 
 sufficient unless we introduce a new force C from outside 
 and make C=T, we shall have horizontal motion." From 
 his failure to state how T balanced C, the writer here loses 
 the thread of his argument and thus reaches a conclusion 
 entirely at variance with the premises laid down by him- 
 self. He should have said, to be consistent, "the pressure 
 of the half beam A against" and opposite the "new force" 
 C must be equal to the ' 'dormant' ' or shearing force T. 
 And as shearing force T, that is equal and opposed to C, is 
 not a "new force" but the same force T in dnew place, he 
 would have reached the conclusion that the load I, has only 
 one horizontal component instead of two, and the moment 
 
 (i) Prof. DuBois, Bng. News, Jan. 2ist, 1888. 
 
12 
 
 of resistance is the moment of a single stress instead of that 
 of a stress couple." 
 
 The writer also erred in giving the ist condition as his 
 reason for making T C to prevent horizontal "traveling", 
 in the whole beam, that is accomplished from the forces 
 complying with our 8th condition, which is solely sufficient, 
 and it was a fatal error to "conceive" that the part could 
 "travel" independently of the whole. His true reason for 
 this is found in our yth condition, in order that T may pos- 
 sess leverage to resist rotation. 
 
 The advocates of the "common theory of flexure" insist 
 that the tension must balance the compression in the beam, 
 just as the force at the supports balances the load, and as 
 their text-books on mechanics demonstrates no other 
 method of balancing except through the aid of an auxiliary 
 shearing force, then they must accept all of the absurdities 
 and contradictions of this method when applied to a solid 
 beam. 
 
 Distribution of the compression. It is rightly claimed by 
 "the -common theory" that at any section the tension is 
 distributed through the portion of the beam that it occu- 
 pies as an uniformly varying force. When the resultant T 
 balances C the laws of mechanics require that the shearing 
 force T shall be greatest at n t Fig. 2, where the common 
 theory requires its opposing force to be zero. The vertical 
 shearing force is a maximum at the inner side of the sup- 
 ports, this horizontal shearing force must follow the same 
 law, and be uniformly distributed along the line ns instead 
 of as an uniformly varying force. Wiesbach states that n 
 is the center of rotation this limits the lever-arm of T to 
 its distance from n, and this will be all of the moment that 
 sustains the load, as it is force T that posseses leverage 
 and not shearing or ' 'dormant' ' force T. 
 
 The neutral line destroyed. The method of balancing T 
 and C, adopted by ' 'the common theory' ' requires that two 
 equal and opposite shearing forces T (or C's after they be- 
 came opposed) shall pass through the neutral line, the ef- 
 
13 
 
 feet of which will be to destroy the properties claimed for 
 the line by the theory. 
 
 Is it not a mechanical absurdity to say that T and C 
 balance each other horizontally as right line forces, when 
 their lines of action do not coincide and that their is be- 
 tween their lines of action a neutral or dead line over which 
 the effect of neither force can pass ? The equal but unbal- 
 anced forces of a "static couple" that simply overcomes 
 
 /\ 
 
 the inertia of the body and causes it to rotate about its cen- 
 ter of mass has what resembles the neutral line, but it dis- 
 appears as soon as the forces balance each other by the 
 bodies transmitting a shearing force. 
 
 A solid beam has a neutral line and the compression 
 varies uniformly, neither of which could exist, as such in 
 the beam, at the same time with a longitudinal shearing 
 force. The tension cannot balance the compression with- 
 out this force, therefore they do not balance, and the theory 
 must be reversed in order to explain the facts. 
 
 Method of Sections, or of Substituting ' * applied' ' for gener- 
 ated Forces. The Railroad Gazette 2 says of my objection 
 to the establishment of the "common theory" by what is 
 known as the "method of sections" that "this mode of pro- 
 cedure is so well understood in the treatment of stress that 
 
 (2) September 6th, 1889. 
 
14 
 
 it seems unnecessary to refer to it unless we have made an 
 error in its application," this error I think is evident. 
 
 1 'The applied forces and horizontal stresses on each side 
 of any section of a loaded beam hold each other in equili- 
 brium" each on its own side of this section. Now if we con- 
 ceive one part of the loaded beam to be removed, at any 
 section, and the existing equilibrium in the part that re- 
 mains to be destroyed, or it is reduced to a rigid body and 
 an unbalanced load, then the horizontal forces that we 
 must "apply" at this section to establish horizontal (right 
 line) and rotary equilibrium, at the same time gives us ' 'a 
 clear case of statics." This is the "well recognized" con- 
 dition of equilibrium when the vertical applied force gener- 
 ates nothing but free rotation or its tendency, and it cer- 
 tainly cannot be identical with its condition when it gener- 
 ates constrained rotation accompanied by horizontal 
 stress through "deformation," therefore the "common the- 
 ory" leaves out of consideration, entirely, the effect of the 
 generation upon the generating cause when it substi- 
 tutes -the right handed applied couple (T C), Fig. i for the 
 left handed generated couple in A, for as mechanical 
 powers they are of entirely reverse action. 
 
 But if we conceive, as we must, that the same equili- 
 brium that existed in the half beam before removal must 
 exist in it, after removal, as we have taken away neither 
 the cause, the "applied" load, nor the effect, the generated 
 stress, on its own side of the section, which equilibrated 
 each other in the whole beam, and therefore must do so in 
 the half beam, then it is not such "a clear case of 
 statics." In each halt of the beam, after being divided, we 
 will have a vertical applied force, whose moment has been 
 exhausted or equilibrated by the moment of its two gener- 
 ated horizontal components, otherwise our cutting and 
 removing will introduce an element that did not exist in 
 the loaded beam as a whole, viz: the unbalanced moment of 
 the vertical load, which we are not warranted in doing for 
 the pleasure of balancing if again in some other manner. 
 
15 
 
 The bending of a beam, under a tranverse load, is actual 
 rotation around some point, and not simply a tendency 
 around every known point, were it not so the stresses could 
 not arrange themselves on each side of the neutral line as 
 as they are known to do. Statics does not recognize that 
 there can be any motion in an equilibrated system of forces 
 by reason of the "deformation" of the body upon which it 
 acts, but conceives that the body cannot bend or break un- 
 der the action of any force, however, intense, but always 
 retains its size and shape unchanged. 
 
 If the rotation of the beam A, Fig. i by the load L,, gen- 
 erated nothing but free rotation or tendency, which is all 
 that is contemplated by statics, then "the common theory" 
 method of balancing by the section method is correct. But 
 the rotation or tendency instead of being free, where every 
 point in the body tends to move in the arc of a circle with 
 a radius equal to its distance from the common center, it is 
 constrained rotation where every point actually moves in a 
 curved line, with a constantly changing distance from ' 'the 
 common center, ' ' from each such point having a longitudi- 
 nal motion. 
 
 In free rotation or its tendency the load generates no 
 stress components as there is no horizontal motion, and its 
 tendency to freely rotate the beam in one direction must be 
 resisted by outside "applied" forces that tend to rotate it, 
 equally, in the opposite direction, as in the common theory, 
 but the constrained rotation of the beam generates two hor- 
 izontal stress components, and the effort expended by the 
 load in their generation equilibrates and absorbs its rotat- 
 ing power, hence there is no unbalanced rotating tendency 
 for our outside "applied" forces, at the section, to balance 
 by rotating the beam in the opposite direction. 
 
 The method of sections will lead to no error if it is sim- 
 ply used to interpret the equilibrium that is established 
 through the generation of horizontal forces instead of es- 
 tablishing it de novo as is done in the "common theory." 
 In this interpretation two separate and distinct uses must 
 
16 
 
 be made of the "applied" forces, ist, to apply such forces 
 that will establish the horizontal equilibrium that existed 
 in the whole beam, 2nd, to apply such forces that their me- 
 chanical power, or moment, will resist the rotating power 
 of the load in the same identical way that it was equilibrat- 
 ed by the moments of the generated stresses on its own side 
 of the section of division, and both objects cannot be ac- 
 complished at the same time by the same diagram of forces 
 as is done in the "common theory." 
 
 The first problem then is to apply at the removed section 
 two such force systems, of which T and C are the result- 
 ant Fig. i, that will enable the half beam A to sustain the 
 load L, with the same generated stresses in A that existed 
 before the removal of the other half of the beam. After 
 division the horizontal stresses must be conceived to exist 
 in each part, just as they did before division, and as it 
 would be impossible for these stresses to import motion to 
 the half beam in which they exist, whether they are equal 
 or not, there is no mechanical necessity for our making 
 our applied forces equal. We must then find only such 
 forces that must have existed in the removed portion of the 
 beam in order that such forces when applied will produce 
 with the forces already in the half beam, the observed 
 horizontal deformation, that is, the elongation by tension 
 and the shortening by compression and the original hori- 
 zontal equilibrium will be restored by applying a pull equal 
 and opposite to the existing pull, and a push equal and op- 
 posite to the existing push, and this is the only condition 
 we can legitimately impose upon these forces, when they 
 are equal opposed pairs and conspiring at any section. 
 "The applied" forces T and C, Fig. i represent in the sys- 
 tem the removed load on B, and thus enables A to sustain 
 the same load that it did before removal and with the same 
 generated stresses in A. 
 
 The second problem then is to find two such forces that 
 when substituted for the left handed couple generated in A, 
 by the load I,, Fig. i, the mechanical action of the substi- 
 
17 
 
 tuted or ' 'applied" and the generated forces will be identi- 
 cal in resisting the rotation of A, produced by L. All of 
 our knowledge of mechanical power is derived from exper- 
 ience and not from a priori reasoning, and we must follow 
 the same teacher when we substitute the moment of an 
 "applied'' force for the moment of a "generated" force. A 
 generated tensile force and the generating force have the 
 same sign, and if both were applied or possessed rotating 
 power they would rotate the beam in the same direction, but 
 a generated compressive force and the generating force have 
 opposite signs and tend to rotate the beam in opposite di- 
 rections. The rule for the substitution then is, the moment 
 of a generated tensile force is the moment of an * 'applied' ' fotce 
 in the opposite direction, and the moment of a generated com- 
 pressive force is the moment of an "applied" force in the 
 same direction. The "common theory" recognizes the cor- 
 rectness of this rule when it substitutes the moment of the 
 "applied" force T, Fig i, for the moment of the opposite 
 generated tensile force in A, but ignores it, and thus the 
 teachings of experience when it substitutes the" moment of 
 the "applied" force C, for the moment of the opposite gen- 
 erated compressive force in A, and thus arrives at an error- 
 nious conclusion. For resisting the rotation of A, it re- 
 quires a force C in the same direction with T. 
 
 Equilibrium at the Section of Maximum Stress. Conceive 
 the beam to be divided, at the section of greatest horizontal 
 stress, and the forces to be balanced in each part as claimed 
 by the common theory, then place them together but sepa- 
 rated by the rubber body as shown in Fig. 3. The lines of 
 action of the (T T) and (C C) forces are represented as not 
 conciding so as not to obscure the action that is claimed for 
 them. 
 
 Now if + T balances C, it can only do so from the 
 body A, transmitting the auxiliary shearing force of -f T, 
 to the line of action of C, and this becomes + C, which 
 is the shearing of -f T, and if T balances + C, it can 
 only do so from the beam B, transmitting the shearing force 
 
18 
 
 ot T, to the line of action of + C, and this becomes C. 
 Then the two forces + C and C are the shearing forces 
 of the pair of T forces and compress the rubber body j ust 
 as if they had been originally applied to it. 
 
 The common theory makes T sustain one-half of the 
 load on B and its shearing force C, one-half of that on 
 A. Now force T, and its shearing C, are in nature 
 and mechanics the same identical force, this theory then 
 makes force T exhaust it power to sustain the load twice 
 over, at the same time, or in other words, it sustains the 
 whole load on either A or B, which is true in this method 
 
 r + T 
 
 A A 
 
 Pis. 3 
 
 of balancing, but it is done in machines directly, without 
 the fiction of giving"the shearing forces leverage. 
 
 The development of the static forces (T T) must be 
 attended by their'equal shearing forces, (C C) in this theory, 
 or they can possess no leverage resistance to the rotation of 
 the loads (L L). The identity of the pair of (T T) forces 
 and the pair of (C C) in this equilibrium follows from the 
 ' 'parallel ogran of forces"; for produce the lines of action of 
 each of the T forces, backward, until each intersects that 
 of its load I/. From these points draw the two diagonals 
 to the points of support and there decompose them and we 
 will obtain the pair of T forces which now become a pair 
 C forces. This also, demonstrates that a neutral line and 
 
an uniformly varying compressive force cannot exist in this 
 condition of equilibrium. 
 
 Resisting Moment. A writer states that at any section, 
 "The moment of the external force or forces will be equal 
 but opposite in sign, to the internal resisting moment." A 
 resistance pure and simple like that offered by a rope to 
 extension or a column to compression can have no resisting 
 moment, nothing but a force can have a moment. The 
 internal resistance offered by the material of the beam is 
 the passive measure of the intensity of the forces. The 
 vertical load generates the horizontal forces through its 
 ability to rotate the beam, the material passively resists this 
 rotation and measures the intensify of the horizontal effort. 
 
 No error would result in calling it a "resisting moment" 
 as a mark of identification, but when it is further said that 
 "the sign of this moment must be opposite to that of the 
 load" that produced it, a fatal error is introduced from an 
 entire misconception of the subject; for it is conceded that 
 the vertical applied forces generate the horizontal forces, 
 each on its own side of the section of maximum stress. 
 
 It is a well recognized principle that like produces like 
 and not unlike, that the energy of any movement or tend- 
 ency is balanced by that which it generates in exhausting 
 itself. Then to require the load on A, Fig. i, to exhaust 
 its rotating power by producing a right handed couple (T 
 C) violates undisputed principles of nature. 
 
 If the load generated nothing but rotation and the balanc- 
 ing "couple" was of entirely different origin the statement 
 would be true. From their symmetrical arrangement the 
 horizontal forces, at the section of maximum stress, has the 
 semblance of this truth, but at any other section it has not 
 even this semblance. 
 
 Equilibrium at Any Section. All writers on the common 
 theory of flexure agree that at any section "the moment of 
 the applied or deflecting forces equals the sum of the 
 moments of the resisting forces. 
 
 Then let us consider the equilibrium, at the quarter sec- 
 
20 
 
 tion of the half beam A, by removing the left hand quarter 
 of the beam A B thus giving us Fig. 4. Between this 
 quarter and the middle section the molecular movement of 
 the material is to the left, above the neutral line, and to 
 the right below this line. The direction of these movements 
 demonstrate that there are only two forces acting between 
 these sections, one pulling the material of the beam from 
 and the other pushing it to the middle section. The forces 
 t and c acting at the quarter section are then represented in 
 direction by the arrows in Fig. 4, and there are no others. 
 
 In the "common theory" the load L on B is balanced by 
 the left-handed couple (T C) at the middle section, and all 
 unbalanced vertical forces being thus provided for, we are 
 lead to inquire from what outside source do the forces t and c 
 come and what do they represent. They are, at this sec- 
 tion the generated components of the load L that rested 
 upon the removed quarter of the beam A B and represent 
 it in the system of forces given in Fig. 4 and the moment 
 of the removed load, though forming apparently a left- 
 hande.d couple must have been balanced, in the whole beam 
 by the left-handed couple (/ c) in defiance of the contrary 
 law of statics they do balance, though in accordance with 
 our rule for substituting "applied" for generated forces; in 
 order to determine the mechanical moments of the generated 
 forces / and c when resisting the moment of the removed 
 load on A, the center of rotation being at the bottom of the 
 section 
 
 In discussing this equilibrium the "common theory" 
 writers represent a right-handed couple as acting at this 
 section, composed of forces directly opposite in direction to 
 / and c, which is a fiction, as we have j List shown that / and 
 c, as represented in Fig. 4, are the only horizontal forces 
 that can possibly act at this section and produce the ' 'de- 
 formation' ' that is known to take place. The fiction that 
 there exists at this section a "resisting moment opposite in 
 sign" composed of the resistance of the material to longi- 
 tudinal tension and compression, will not do, for this resist- 
 
21 
 
 ance of the material simply measures the horizontal intenA 
 sity of / and <:, and being passive, from the very definition 
 can have no moment or tendency to rotate A. One class of 
 writers 3 represent a right-handed "strain couple" and an- 
 other 4 a right-handed * 'force couple" as acting at this quar- 
 ter section, when in fact it is left handed apparently. 
 
 The forces / and c cannot balance each other, and, there- 
 fore, cannot be generated equal at this section, although it 
 is a fundamental principle of the common theory that they 
 should. 
 
 We have seen that in order that c shall be balanced by t, 
 the beam A must transfer t as a shearing force until it be- 
 comes opposite c on its own line of action; this it cannot do 
 
 Fie. 4 
 
 without destroying the equality that is claimed to exist be- 
 tween the tension and compression at the middle section, 
 for with a shearing force t acting opposite to c, at the quar- 
 ter section, the resultant compression at the middle section 
 will be C / as the compressive and shearing force act in 
 opposite directions along the same line of action, the re- 
 sultant force C t will therefore be less than T. 
 
 Then as no shearing force can act in the opposite direc- 
 tion to that of r, at this or any other section, without dim- 
 inishing the compression on the middle section, these tension 
 and compression forces cannot balance at the middle or sec- 
 tion of maximum stress. For these horizontal T and C forces 
 
 (3) Weisbach's Mechanics of Eng., pp. 548 (Coxe). 
 
 (4) Wood's Elements of Analytical Mechanics, pp. 129. 
 
22 
 
 increase from zero at the ends to their maximum value at the 
 middle section of the beam. When the lever- arm of the 
 load is zero the horizontal forces are zero, then each incre- 
 ment to the lever- arm of L, gives an increment to the value 
 of T and C and if the forces cannot balance and develope 
 equally at the section of their origin, the opportunity is lost 
 and this condition cannot be imposed upon them after they 
 come into existence, and this they cannot do, for it requires 
 at each such section the horizontal effect of at least three 
 forces to balance, which can only be fulfilled at the middle 
 section. 
 
 Spring Experiments. Certain experiments made by 
 cutting the beam into two pieces and reconnecting the parts 
 by placing elastic springs between them and opposite the 
 resultants of the tension and compression of Fig. 3, thus 
 giving it the semblance of the original, and the develope- 
 ment of equal tension and compression in the springs has 
 caused some minds to conclude that the demonstration of 
 the common theory "has all the absoluteness of geometry.'' 
 
 Prof. Rankine says of the right line balance of parallel 
 forces when applied to a body, "that all pairs of directly 
 opposed equal forces may be left out of consideration, for 
 each such pair is independently balanced what soever its 
 position may be." The horizontal forces in the whole beam 
 consists of two such directly opposed pairs, and the body 
 as a whole must remain at rest whether the forces of these 
 pairs are equal or not; for there is no unbalanced force to 
 cause horizontal motion. Now, if the whole is at rest with- 
 out the forces of these paris being equal, must not the half 
 of the whole be in the same condition? Then "the com- 
 mon theory" equality of the tension and compression in the 
 beam, and its semblance, when its parts are connected by 
 springs, cannot be a necessity from the first condition of 
 equilibrium, but results from an entirely different mechan- 
 ical principle. 
 
 Moment is the result of the arrest of rotary motion 
 when we pull a rope with a force at each end right line 
 
23 
 
 motion is arrested but neither force possesses any leverage, 
 but should these equal and opposite tensile forces be devel- 
 oped in arresting rotary motion, each would possess lever- 
 age in resisting the cause of the rotation in proportion to 
 their distance from the center. This leverage it can only 
 have from the bodys transmitting a shearing force to the 
 center of rotation. 
 
 In the spring experiments the tension spring arrests the 
 rotary motion and each half-beam transmits to the compres- 
 sion springs, the centre of rotation, equal and opposite 
 shearing forces (in order to possess leverage) which causes 
 the compression to develope equal to the tension, which is 
 not a result of the first condition. 
 
 The half beams A and B, Fig. 3, connected by springs 
 possess another property that is fatal to the supposed iden- 
 tity between it and the solid beam. After placing a pair of 
 springs at the middle, divide the half beam into two parts 
 at any section and reconnect them by means of another pair 
 of springs on the same horizontal line with the first pair, 
 then demonstration and experiment shows that the stress 
 in each pair is the same, thus making the tension and com- 
 pression uniform, from end to end, in the common theory 
 beam, when in the beam as it is, they are zero [at the ends 
 and a maximum at some interior section. 
 
 This uniformity of stress, from end to end of the beam, 
 in the * 'common theory," results from one of the oldest 
 known principles of statics, "the parallelogram forces." 
 In Fig. i, prolong the lines of the forces Land T until they 
 intersect, then the resultant would move A in the direction 
 of its diagonal, which must be resisted by decomposing the 
 resultant into its original components JL, and T, and apply 
 C and S equal and directly opposite and equilibrium will 
 be established. If we decompose the resultant of L, and T 
 at any point on the diagonal of A, we will obtain the orig- 
 inal components; that is, the tension T does not vary in 
 value. If we now make the same diagram of forces foi the 
 right hand half of the beam B, and place them together as 
 
24 
 
 in Fig. 3, with a spring opposite to the line of action of the 
 tensile forces and remove the applied T forces, then by ap- 
 plying the loads Iy to both A and B, the upper ends of the 
 diagonals, being now connected, they will generate an uni- 
 form tensile stress between these points. And there can be 
 no equilibrium of moments between the deflecting and 
 resisting forces at any point in the beam, except at the sec- 
 tion of maximum stress. 
 
 This failure of the stresses in "the common theory" 
 beam to decrease longitudinally from the section of maxi- 
 mum stress, results from another "well recognized" princi- 
 ple of statics. An applied force distributes itself through 
 the body on its line of action without increase or decrease, 
 and its point of application may be taken anywhere on this 
 line in the body. The applied forces T and C, Fig. i, 
 must follow this law and be of uniform tensity from end to 
 end of A. 
 
 Here again it is seen that the common theory adopts a 
 process that leads to results that are entirely the reverse of 
 what is true for the beam. 
 
 Center of Rotation. From moments measuring the tend- 
 ency of forces to produce rotation about a point, there must 
 be, in every body that is acted upon by a system of forces 
 that balance through the equality of their moments, a com- 
 mon center of such tendency to rotation, which determines 
 the lengths of the lever-arms of the forces. And if the 
 equilibrium of this system is thoroughly understood, the 
 amount and character of the work performed by each indi- 
 vidual force must be clearly ascertained to know the tool 
 that it works with, its lever-arm or distance from the cenier 
 of rotation must be accurately determined. If the line of 
 direction of a force passes through the center of rotation, 
 the lenth of its lever-arm is zero, then by giving it a mo- 
 ment about some other point, we give it what it does not 
 possess. This is the condition in the equilibrium of the 
 common theory, the horizontal forces develop equal, from 
 the pressure producing forces passing through the center of 
 
25 
 
 rotation. When this is the case ' 'we may take our origin 
 of moments anywhere, ' ' but the results obtained express a 
 mathematical play instead of a mechanical condition. 
 
 An algbraic equation may express the equilibrium as an 
 arithmetical problem, but entirely misrepresent the mechan- 
 ical condition. Had the common theory writers determined 
 the actual center of rotation and formed their equation to 
 express the mechanical conditions about this center, instead 
 of as the R. R. Gazette* says as an apology for not doing 
 so, the arithmetical results for the couple being ' 'constant 
 wherever be the origin of moments," which "shows that it 
 is not necessary to assign the origin of moments, ' ' it would 
 have condemned and shown the absurdities of the theory. 
 To obtain this constant arithmetical result, when the origin 
 is between the lines of action of the forces of the couple, 
 their moments, being of the same sign, add; but when the 
 center is outside, the moments of the, forces must subtract, 
 as they then have opposite signs, in order to maintain this 
 "constant" arithmetical result. It does appear very im- 
 portant that we determine by locating the center of rotation, 
 whether the pair of forces work together or work against 
 each other. 
 
 The center of rotation has been variously located by dif- 
 ferent writers. Wiesbach says it is on the neutral line in 
 the section of maximum stress. Another writer, 5 after locat- 
 ing it at the center of A figure i, says it may be "any- 
 where ;" for the couples (L S) and (T C) "tend to turn the 
 half beam around its center of mass. Equilibrium being 
 established, the tendency to turn about any point within 
 or without the beam is zero, and the origin of moment may 
 be anywhere." The motion itself may be zero, but the ten- 
 dency can never be zero while the forces possess leverage. 
 Therefore our writer must let it remain at the center of 
 mass of A, as it cannot be removed by bad logic from where 
 worse mechanics located it. 
 
 (5) R. R. Gazette, September 6th, 1889. 
 
26 
 
 Another author 6 says : "The usual method of failure of 
 beams is by cross-breaking, this is caused by the external 
 forces producing rotation around some point in the section 
 of failure ' ' ; but fails to locate this very important point. 
 
 In a purely static system of forces, such as the ' ' common 
 theory," the center of rotation is easily ascertained, for the 
 shearing of each force travels directly to this point, and 
 the intersection of their lines of travel locates it unerringly. 
 
 In the "spring experiments," and all other devices by 
 which it is demonstrated that the tension and compression 
 develop equal, the compressed spring, or whatever repre- 
 sents it, is, and must be, for this equality, the center of ro- 
 tation or tendency. Then in the solid beam, the beam must 
 be able to rotate around a point on the compressive force, 
 which it cannot and has no appearance of doing during 
 bending, therefore the . compression and tension cannot be 
 generated equal in the solid beam. 
 
 But in the beam as it is, we have only a vertical shear- 
 ing force, which alone is not sufficient to locate the 
 center,, and we must combine with this the knowledge de- 
 rived from the character of the ' ' deformation ' ' of the beam 
 during bending. The vertical shearing forces locates the 
 center of rotation in the section of maximum stress, and as 
 this is also the only plane that remains " fixed," relatively, 
 to the whole, during the process of bending, the center of 
 rotation must be a point of this plane as it is a * ' fixed 
 point" also. 
 
 If we divide a beam, that is supported at its middle and 
 loaded at its ends, into very thin horizontal layers, each 
 distinguished from the other by differences in color, then 
 during the process of bending the points of each layer will 
 be observed to have a downward motion, and that the 
 points of no layer can rotate around a point, as a center, in 
 a plane higher than its own the points of the bottom layer 
 can rotate around a point in its own plane, but not around 
 
 (6) Merriman's Mechanics of Materials, pp. 42. 
 
27 
 
 a point in a plane higher than its own ; then, as there can 
 only be one such point for the whole beam, it must be at 
 the intersection of the plane of maximum stress with the 
 plane of the concave or compressed side of the beam, where 
 Galileo determined it to be nearly two and a half centuries 
 ago. 
 
 Development of Equilibrium. When an unbalanced force, 
 such as a vertical load, is applid to a beam or body, and 
 equilibrium is established either by the development or the 
 application of forces acting at right angles to the direction 
 of the applied, unbalanced force, two cases may be dis- 
 tinguished : 
 
 1 . The resultant moment is the moment of a single force 
 in the direction of each axis, or the compressive forces have 
 no moment, as their lines of action pass through the center 
 of rotation or tendency. 
 
 2. The resultant moment in the direction of the axis of 
 the applied force only, is the moment of a single force, and 
 in the direction of the other it is the sum of the moments of 
 two components, or the compressive component has leverage 
 as well as the tension, and neither have any shearing force. 
 These distinctions the writer has never seen made by any 
 text book of mechanics. When motion could take place 
 in the body, or half body, as a whole, were it not for the 
 conditions of ' ' fixedness ' ' imposed by connections outside 
 of the body itself, then the arrest of the rotary motion of 
 the body, as a whole, produces tension in the "fixing" 
 farthest from the center of rotation and equilibrium of mo- 
 ments is established from both the applied force and the 
 developed tension transmitting their shearing forces to the 
 center of rotation. 
 
 In Fig. i we have the load I/, the reaction of the support 
 S, and equal and opposed to S the shearing force L,, thus 
 giving us the vertical effect of three forces, the resultant 
 moment of which is that of the single force I,, for as both 
 the opposed forces S and shearing I, pass through the cen- 
 
28 
 
 ter of rotation, their lever-arm is zero, consequently the mo- 
 ment of the three is that of L. An analysis of the equili- 
 brium of the forces acting in the direction of the other axis 
 or the (T C) forces leads to the same conclusion, for if they 
 balance they do not dc so in any mysterious mythical way 
 This principle is the basis of the equilibrium in the spring 
 experiments, nut-crackers, the "common theory" and a 
 host of other contrivances with which we are familiar. 
 
 But when the condition of "fixednes" is such as exists 
 between the two halves of a beam, and each half is solicited 
 to rotate in opposite directions around a fixed center, no 
 motion can take place except through the "deformation" 
 of the beam itself. This requires that its rigedness shall 
 first be overcome by longitudinal compression in order 
 that the half-beam may assume the arc form first by the 
 shorting of the material on the concave side, the result of 
 which is bending and the develoyment of tension. The 
 pressure components are first generated and are not the 
 shearings of the tensile forces, they do not pass through 
 the center of rotation and therefore possess leverage. 
 
 With equilibrium established for any given load L, 
 under our first case, the order of development is as follows: 
 An increment to the load L, gives an increment to the rota- 
 tion, its conversion into a tendency gives an increment to 
 the tension, this in turn gives an increment to its shearing 
 and this an increment to the moment of the tensile force by 
 which equilibrium is re-established. 
 
 In the second case an increment to the load I, gives an 
 increment to the compression which gives an increment to 
 the shortening, this admits of an increment to the bending 
 (which is actual rotation and not a tendency}, and this 
 finally gives an increment to the tension, and equilibrium 
 is re-established by the increments to the horizontal forces 
 giving an increment to the sum of their moments without 
 the increments balancing as right line forces. 
 
29 
 
 THE WRITER'S THEORY. 
 
 The method of generating the horizontal forces, distin- 
 guished in our second case, has been made the basis of the 
 theory given in my book on "The Strength of Beams and 
 Columns" 7 The Railroad Gazettee of July 26, 1889, in its 
 review of this book says: "If a theory containing a me- 
 chanical absurdity leads to rational results, it does not 
 prove that the "fallacy" is truth, but rather that some 
 other error must be "involved to offset the first error." 
 The Gazette then proceeds to demonstrate the existence of 
 the fallacy from glittering generalties and "mathematical 
 necessities" drawn from the formulated principles of pure 
 statics that are applicable, solely and alone, to the first case 
 above given, with equal propriety, its fallacy might have 
 been demonstrated from its failure to comply with "the 
 well recognized principles" of Dynamics, Hydrostatics or 
 those of any other kindred science. This method is cer- 
 tainly not within the scope of fair criticism. 
 
 The horizontal forces do not exist of themselves, neither 
 do we apply them; they are generated. An unloaded beam 
 has none, and if the load be applied pound by pound, the 
 horizontal forces grow and strengthen from zero, with the 
 increase in the load, and the method of their generation is 
 the vital part of any theory of \hzflexure of beams and col- 
 umns. 
 
 If the method of generation, described in my theory, is 
 impossible mechanics, it can be demonstrated, per se, and 
 not because there is another rational method that is appli- 
 cable to entirely different mechanical conditions, and the 
 conclusions drawn from the results of one Iprocess are at 
 variance with those from the other. 
 
 No "common theory" writer, with whom I am familiar, 
 makes the process of generating the horizontal forces any 
 part of his demonstration. The conclusions are "mathe- 
 
 (7) E. & F. N. Spon, publishers, New York and London. 
 
30 
 
 matical necessities," drawn from general algebraic equa- 
 tions expressing possible equilibrium about every point in 
 the universe, and the physical condition of the beam is 
 only considered when the amount of the stresses are needed 
 to give value to the forces of the general equation. 
 
 The Gazette again says. "Many results found by the 
 author agree fairly with those of direct experiment, and in 
 some cases it may give better results than those found by 
 the common theory." Exact identity between the compu- 
 ted strength, from my theor}', and the determined experi- 
 mental strength could not be expected from the record of 
 experiments already made, which was the only source I 
 had from which to draw my illustrations. The tensile and 
 compressive strength, as recorded, is usually the mean of 
 many tests, the single test varying in value from five to ten 
 per cent, from the mean values, while the computed tran- 
 severse strength is for a single test, which test strength may 
 vary from the strength computed from mean values five to 
 ten per cent. Had the number of tests been the same to 
 determine the tensile, compressive and transverse strength 
 then the computed strength, from the mean of the tensile 
 and compressive tests, would be identical with the mean of 
 the transverse tests. In all cases where the strength of the 
 beam was computed from the tested tensile and compressive 
 strength of the identical material contained in the beam 
 broken, the discrepency was three per cent, and less, this is 
 especially noticeable in the very careful experiments of Mr. 
 Kirkaldy and those made at the Watertown Arsenal. 
 
 Columns. The amount of the load applied to a column 
 directly measures the amount of the internal stress, and 
 their relation is the crucial test of the correctness of any 
 theory of flexure. The "common theory" has utterly 
 failed to establish any such relation, Gordon's, the most 
 successful formula, is based upon a direct denial of the cor- 
 rectness of the theory; in its demonstration, it is said, 8 
 
 (8) Burr's Materials of Engineering, pp. 430. 
 
31 
 
 'The condition of stress in any normal section of a long 
 column is that of a uniformly varying system composed of 
 a uniform stress and a stress couple, ' ' the resultant com- 
 pression of this ' 'system' ' is then greater than the resultant 
 tension, which moves the neutral line from the center of 
 gravity of the section. The writers' s theory fully estab- 
 lishes this relation, the correctness of which is amply sus- 
 tained by all recorded experiments. 
 
 R. H. COUSINS. 
 Austin, Texas, August 29, 1891. 
 
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