From inspection of Fig. 19, it will appear evident that, as the position of the truss to the left of the section is in equilibrium, the following statements are true: 1. The algebraic sum of the moments of the exterior forces and the stresses in the members cut by the section, is equal to zero. This is true of the moments taken about any or all points; for, if it were not, the portion of the truss would begin to rotate about some point, and would continue until equilibrium was established.
2. In a vertical plane, the algebraic sum of the components of the exterior forces and the stresses in the members cut by the section is equal to zero; for, if such were not the case, the portion of the truss shown would move up or down with a constantacceleration.
3. The algebraic sum of the components of the exterior forces and the stresses in the members cut by the section in a horizontal plane, is equal to zero; for, if such were not the case, the portion of the truss would move either to the right or to the left, with a constant acceleration.
4. From 2 and 3, above, it is evident that the algebraic sum of the components of the exterior forces and the stresses in the members cut by the section is equal to zero in any and all planes.
The section is not necessarily required to be a vertical line as in Fig. 19. It may be oblique, as in Fig. 20; or it may be a circular section, as shown in Fig. 20a. When the latter is the case, it is said that the sum of the components of the forces around the point U, is in equilibrium in any plane that may be taken.
It is also evident that the forces in the members cut by the section, and the exterior forces to the right, are in equilibrium. This condition is very seldom utilized in the determination of stresses, as that portion of the truss to the left of the section is almost always considered.
28. Resolution of Forces. This method is one of the simplest and at the same time least laborious. The forces are generally resolved into their horizontal and vertical components, or parallel and perpendicular to some member. In cases where two unknown stresses occur, two equations can usually be formed, and these solved. It should be assumed that the unknown stress acts away from the section which cuts it. It will then solve out, with the proper
sign indicating the character of the stress—that is, whether it is tensile or compressive. Tensile stresses are indicated by placing the plus (+) sign before them, while a minus (—) sign indicates compression.
A few equations showing the application of the method of the resolution of forces can be written after inspection of Figs. 21 to 25 inclusive. In all cases, S. is the unknown stress, and is assumed to be acting away from the section. The other stresses S„ etc., are known, and their direction given them accordingly, it toward the section if the member is in compression, and away from the section if the member is in tension. Forces or components acting upward or to the right are considered plus; those acting down ward or to the left are considered minus. For a fuller explanation, see the instruction paper on Statics, Articles 17 to 23 inclusive.
In Fig 21, the sum of the vertical components is taken, and the equation is: whence, In Fig. 22, the section is oblique, and the sum of the vertical components is taken: • whence.
In both of the above cases, it will be .noted that the chord stresses do not enter into the equation, as their vertical components are zero.
In Fig. 23, the sum of the horizontal forces is used in deter mining the stress Note that the exterior forces It and p do not enter the equation, as they are not to the left of the section, and also their horizontal components are zero.
whence, in Fig. 24, the sum of the vertical forces is again used. Here the section cuts the member with the known tensile stress whence, In Fig. 25, use is made of the fact that the sum of the components of the forces about a point is zero when they are resolved in any plane. IIere they will be resolved perpendicular to the diagonal.
These are some of the most common cases which occur in the determination of stresses in simple trusses. In all cases, follow this method of procedure: 1. Pass a section cutting as few members as possible, one of which must be the one whose stress is 2. The stress in all the members cut, with but one exception, must be known.
3. Write your equation, always placing it equal to zero.