Essentials in the theory of framed structures . THEORY OF FRAMED STRUCTURES Chap. VI span, to which the diagram PUSV is added to provide forcontinuity. Elh = Pk{i - k)li-^ i\- kl + ^(i - k)l + Mi{l l\(- l) 6 3 £7^3 = Mi(- ai\(-al\ + Ma/- al\(- al\ = ^(2Mi + M2)0 Elh = mJ- ai\(- al\ + Mil- al\(- al\ = ^(Mi + 2M2)o Whence Mi = \ , „ , 3^2 + 8a + 4 Fl{k - k^)a3a2 + 8ff + 4 Ri = P(i - /fe) - 2P(A! - k){a+ i)3^^ + 8a + 4 R,=Pk+ P(k - k){2a+ 5a + 2)isa^ + 8a + 4)a j;^_ -Pik-k){a + 3a+2)(30^ + 8a + 4)a /?4 = P(/fe - <fe^)a3a^ + 8a +4 168. Three Equal Spans—^Unifonn Load.—The beam in is con
Essentials in the theory of framed structures . THEORY OF FRAMED STRUCTURES Chap. VI span, to which the diagram PUSV is added to provide forcontinuity. Elh = Pk{i - k)li-^ i\- kl + ^(i - k)l + Mi{l l\(- l) 6 3 £7^3 = Mi(- ai\(-al\ + Ma/- al\(- al\ = ^(2Mi + M2)0 Elh = mJ- ai\(- al\ + Mil- al\(- al\ = ^(Mi + 2M2)o Whence Mi = \ , „ , 3^2 + 8a + 4 Fl{k - k^)a3a2 + 8ff + 4 Ri = P(i - /fe) - 2P(A! - k){a+ i)3^^ + 8a + 4 R,=Pk+ P(k - k){2a+ 5a + 2)isa^ + 8a + 4)a j;^_ -Pik-k){a + 3a+2)(30^ + 8a + 4)a /?4 = P(/fe - <fe^)a3a^ + 8a +4 168. Three Equal Spans—^Unifonn Load.—The beam in is continuous over three equal spans /, and supports a uni-form load of w lb. per unit of length. Since the beam is sym-metrical about the center, the ordinates in the M-diagram atB and C are equal; and only one elastic equation is the tangent to the elastic curve be drawn through B, and Sec. II RESTRAINED AND CONTINUOUS BEAMS 26s let h and ^3 represent the tangential deviations at A and Crespectively; then ^ if Ml) ^<m 24 3. Fig. 165. Eltz 8 rn^n+Mj 0) _ wl^ MP24 2/i = ~h ^ + MI1= _f1 _ Ml24 3 24 2 M = 10 i?i = i24 = — wl10 i22 = -K3 = — z*^10 ? If the M-diagram had not been symmetrical, either onaccount of unsymmetrical loading or variation in span lengths, 266 THEORY OF FRAMED STRUCTURES Chap. VI or both; the ordinates at B and C could not have been assumedequal. In this case there would have been two unknownordinates, M2 at B and Ms at C; and two elastic equations wouldhave been necessary for a solution. These two equations may be written by establishing a rela-tion between h and tz, and a relation between h and h; or onetangent may be drawn through B and another through C. 169. Four Equal Spans—Uniform Load.—The beam in is continuous over four equal spans I, and supports auniform load of w lb. per unit of length. Since the beam issymmetrical about the center, there are only two unknownordinates Mi and M3 to be determined, for which two ela
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