i  fM (5.153)  M j j M Mi i    k   k (5.154) Mj j The flexibility matrix f then will be a 2
2 matrix. The first column can be obtained by setting Mi  1 and Mj  0 (Fig. 5.92b). The resulting angular rotations are given by Eqs. (5.107) and (5.108): For a beam with constant moment of inertia I and modulus of elasticity E, the rotations are   L/3EI and   L/6EI. Similarly, the second column can be developed by setting Mi  0 and Mj  1. The flexibility matrix for a beam in bending then is  3EI 6EI L 2 1 f   The stiffness matrix, obtained in a similar manner or by inversion of f, is L L 2EI 2 1 k   Beams Subjected to Bending and Axial Forces. For a beam subjected to nodal moments Mi and Mj and axial forces P, flexibility and stiffness are represented by 3
3 matrices. The load-displacement relations for a beam of span L, constant moment of inertia I, modulus of elasticity E, and cross-sectional area A are given by  Mi Mi i   f M M  k  (5.157)  j  j  j  j e P P e In this case, the flexibility matrix is

To find reaction R1, we take moments about R2 and equate the sum of the moments to zero (clockwise rotation is considered positive, counterclockwise, negative): R  14,000 lb 1 In this calculation, the moment of the uniform load was found by taking the moment of its resultant, which acts at the center of the beam. To find R2, we can either take moments about R1 or use the equation V  0. It is generally preferable to apply the moment equation and use the other equation as a check. R  13,000 lb 2 As a check, we note that the sum of the reactions must equal the total applied load: 5.5.3 Internal Forces Since a beam is in equilibrium under the forces applied to it, it is evident that at every section internal forces are acting to prevent motion. For example, suppose we cut the beam in Fig. 5.17 vertically just to the right of its center. If we total the external forces, including the reaction, to the left of this cut (see Fig. 5.18a), we find there is an unbalanced downward load of 4000 lb. Evidently, at the cut section, an upward-acting internal force of 4000 lb must be present to maintain equilibrium. Again, if we take moments of the external forces about the section, we find an unbalanced moment of 54,000 ft-lb. So there must be an internal moment of 54,000 ft-lb acting to maintain equilibrium. This internal, or resisting, moment is produced by a couple consisting of a force C acting on the top part of the beam and an equal but opposite force T acting on FIGURE 5.18 Portions of a beam are held in equilibrium by internal

in sound level, dB Change in apparent loudness 3 Just perceptible 5 Clearly noticeable 10 Twice as loud (or 1/2) 20 Much louder (or quieter) p SPL  20 log (11.13) 10 po Sound power level refers to the power of a sound source relative to a reference power of 1012 W. (Note: At one time, 1013 W was used; thus, it is imperative that the reference level always be explicitly stated.) The ear responds in a roughly logarithmic manner to changes in stimulus intensity, but approximately as shown in Table 11.25. Another comparison, which gives more meaning to various levels, is shown in Table 11.26. Measurement Scales. Most measurements or evaluations of sound intensity or level are made with an electronic instrument that measures the sound pressure. The instrument is calibrated to read pressure levels in decibels (rather than volts). It can measure the overall sound pressure level throughout a frequency range of about 20 to 20,000 Hz, or within narrow frequency bands (such as an octave, third-octave, or even narrower ranges). Usually, the sound-level meter contains filters and circuitry to bias the readings so that the instrument responds more like the human eardeaf to low frequencies and most sensitive to the midfrequencies (from about 500 to 5000 Hz). Such readings are called A-scale readings. Most noise level readings (and, unless otherwise specifically stated, most sound pressure levels with no stated qualifications) are A-scale readings (often expressed as dBA). This means that actual sound pressure readings have been modified electrically within the instrument to give a readout corresponding somewhat to the ears response (Fig. For various measurements and evaluations of performance for materials, constructions, systems, and spaces, see Art. 11.81. This process consists of: 1. Acoustical analysis a. Determining the use of the structurethe subjective needs b. Establishing the desirable acoustical environment in each usable area c. Determining noise and vibration sources inside and outside the structure d. Studying the location and orientation of the structure and its interior spaces with regard to noise and noise sources 2. Acoustical design a. Designing shapes, areas, volumes, and surfaces to accomplish what the analysis