4. Solve for your stress.
29. Method of Moments. The stresses in all members of a truss can be determined by this method. By section 1 of Art. 28, the point about which the moments are considered can be taken anywhere. Fig. 26 represents the point as taken somewhere outside of the truss at"a distance a above the point U,. The equation will then be: This involves three unknown quantities, and therefore two other points should be taken, and two more equations written. By the use of the three equations, the stresses can be determined.
It is customary to assume the center of moments at such a place that the moments of all the un known stresses, with one excep tion, are zero. This condition requires that their lines of action pass through the center of mo ments. Let it be required to determine the stress If the center of moments is taken at then, as the lines of action of S, and S, pass through this point, their moments will be zero, and the following is true: +R X 2p P, X p f P, X O X ti = O.
Likewise, if the top chord is curved, the center of moments can be taken in such a position that only the unknown stress will enter into the equation. If it is desired to determine the stress S,, Fig. 27, the equation would be: S, X l R X a + P, (a+ p) + P, (a + 2p) =0, the center of moment being at0, the intersection of the lines of stress of and S,. Solving the equation just stated, S, = Ra + P, (a_+ p) + P, (a + 2p) 30. Stresses in Web Members. By reference to Articles 28 and 29, it is seen that several methods are presented for the solution of stresses in web members. Each should be adapted to the case in hand. The simplest method, and the one which is commonly used in all trusses with parallel chords, is by the resolution of the vertical forces. Fig. 21 is to be referred to. The equation given on page 19 is: +R p p S. cos But R p p is equal to V, the vertical shear at the section, and so the equation may now be written: V S, cos whence the following important rule is deduced: The algebraic sum of the vertical shear at the section and the ver tical components of the stress in all of the members cut by the section, is equal to zero.
In trusses with horizontal chords and a simple system of webbing, the equation may be put in the form: S. + V sec 0; and the statement that the stress in any web member is equal to the shear times the secant of the angle that it makes with the vertical is true. The practice of using this latter statement is not to be en couraged, as it leads to confusion in the signs of the stresses. Equa tion (1) should be written in all cases, and the stress will then solve with its correct characteristic sign, indicating that the stress is either tensile or compressive.
As an example, let it be required to determine the stresses in the web members S, and 5, of the Pratt truss shown in Fig. 28, the loads being in thousands of pounds. First, a section should be passed, cutting that member and as few others as possible. Next, the shear at that section should be computed. Then the vertical components of all the stresses cut by the section, and the vertical shear, should be equated to zero. Finally, solve the equation. Remember that the unknown stress is to be assumed as acting away from the section, and that forces or resultants acting downward are considered negative, while those acting upward are considered positive.
To determine S,: The vertical shear at the section u a is: +37.5-2 X 10 5 = +12.5.
As the chord stresses do not exert a vertical component, the equation is: +12.5 + S, = 0 = 12.5, which is a compressive stress of 12,500 pounds.
Note that in this case the angle which the member makes with the vertical is zero, and the cosine and secant are unity.
To determine S,: The vertical shear at the section bb is +37.5-2X10-2X5+7.5.
The equation is: +7.5S, cos ’=0 S, = + 7.5 sec 56.
Sec ’ is equal to 1 30 + _ 30, which is equal to 1.302; and therefore, 8, = +7.5 X 1.302 = +9.765, which is a tensile stress of 9,765 pounds.