deviation s, psi Average strength
cr psi Under 300  400
c 300400  550
c 400500  700
c 500600  900
c Over 600  1200
c TABLE 9.2 Required Average Compressive Strength When Data Are Not Available to Establish a Standard Deviation Specified compressive strength, , psi
c Required average compressive strength, c
r, psi Less than 3000  1000
c 3000 to 5000  1200
c 5000 to 10,000*  1400
c Over 10,000 to 15,000*  1800
c *From ACI 301 Standard Specifications for Structural Concrete. The values for c
r in Table 9.1 are the larger of the values calculated from Eqs. (9.3) and (9.4).


 1.34 ks (9.3) cr c


 2.23 ks  500 (9.4) cr c where k
1.00 for 30 tests, 1.03 for 25, 1.08 for 20, and 1.16 for 15. For an established supplier of concrete, it is very important to be able to document the value of s. This value is based on a statistical analysis in which Eq. (9.1) is applied to at least 30 consecutive tests, and Eq. (9.2) is applied to two groups of consecutive tests totaling at least 30 tests. These tests must represent similar materials and conditions of control not stricter than those to be applied to the proposed project. The lower the value of s obtained from the tests, the closer the average strength is permitted to be to the specified strength. A supplier is thus furnished an economic incentive, lower cementitious materials content, to develop a record of good control (low s). A supplier who does maintain such a record can, in addition, avoid the expenses of trial batches. When no such production record exists, the required average strength
, can cr be determined from Table 9.2. Documentation of the required average strength must be established. The documentation should consist of field strength records or trial mixtures confirming that the proposed concrete proportions will produce an average compressive strength equal to or greater than
. Alternatively, when an acceptable cr TABLE 9.3 Required Air Entrainment in Concrete Exposed to Freezing and Thawing

If the mass with weight W, lb, in Fig. 5.108 is isolated, as shown in Fig. 5.108e it will be in dynamic equilibrium under the action of the spring force  ky and the inertia force (d2y/ dt 2)(W/g). Hence, the equation of motion is W d2y  ky  0 (5.235) g dt 2 where y  displacement of mass, in, measured from rest position. Equation (5.235) may be written in the more convenient form d2y kg d2y  y   2y  0 (5.236) dt 2 W dt 2 The solution is y  A sin t  B cos t (5.237) where A and B are constants to be determined from initial conditions of the system, and kg is the natural circular frequency, rad/ s. The motion defined by Eq. (5.237) is harmonic. Its natural period, s, is T   2  (5.239)  kg Its natural frequency, Hz, is 1 1 kg If, at time t  0, the mass has an initial displacement y0 and velocity v0, substitution in Eq. (5.237) yields A  v0 / and B  y0. Hence, at any time t, the mass is completely located by vy  0 sin t  y cos t (5.241) 0  The stress in the spring can be computed from the displacement y. Vibrations of Lumped Masses. In multiple-degree systems, an independent differential equation of motion can be written for each degree of freedom. Thus, in an N-degree system with N masses, weighing W1, W2, . . . , WN lb, and N2 springs with constants krj (r  1, 2, . . . , N; j  1, 2, . . . , N), there are N equations of

A truss is a coplanar system of structural members joined together at their ends to form a stable framework. If small changes in the lengths of the members due to loads are neglected, the relative positions of the joints cannot change. 5.9.1 Characteristics of Trusses Three bars pinned together to form a triangle represents the simplest type of truss. Some of the more common types of roof trusses are shown in Fig. 6.46. The top members are called the upper chord; the bottom members, the lower chord; and the verticals and diagonals web members. The purpose of roof trusses is to act like big beams, to support the roof covering over long spans. They not only have to carry their own weight and the weight of the roofing and roof beams, or purlins, but cranes, wind loads, snow loads, suspended ceilings, and equipment, and a live load to take care of construction, maintenance, and repair loading. These loads are applied at the intersection of the members, or panel points, so that the members will be subjected principally to axial stressestension or compression. Methods of computing stresses in trusses are presented in Arts. 5.9.3 and 5.9.4. A method of computing truss deflections is described in Art. 5.10.4. 5.9.2 Bows Notation For simple designation of loads and stresses, capital letters are placed in the spaces between truss members and between forces. Each member and load is then designated by the letters on opposite sides of it. For example, in Fig. 5.47a, the upper chord members are AF, BH, CJ, and DL. The loads are AB, BC, and CD, and the reactions are EA and DE. Stresses in the members generally are designated by the same letters but in lowercase. 5.9.3 Method of Joints A useful method for determining the stresses in truss members is to select sections that isolate the joints one at a time and then apply the laws of equilibrium to each. FIGURE 5.46 Common types of roof trusses. Considering the stresses in the cut members as external forces, the sum of the horizontal components of the forces acting at a joint must be zero, and so must be the sum of the vertical components. Since the lines of action of all the forces are known, we can therefore compute two unknown magnitudes at each joint by this method. The procedure is to start at a joint that has only two unknowns (generally at the support) and then, as stresses in members are determined, analyze successive