Ingeniero

BEAM DIAGRAMS AND FORMULAS

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BEAM DIAGRAMS AND FORMULAS Nomenclature

E = modulus of elasticity of steel at 29,000 ksi I = moment of inertia of beam (in.4) L = total length of beam between reaction points (ft) Mmax = maximum moment (kip-in.) M1 = maximum moment in left section of beam (kip-in.) M2 = maximum moment in right section of beam (kip-in.) M3 = maximum positive moment inbeam with combined end moment conditions (kip-in.) Mx = moment at distance x from end of beam (kip-in.) P = concentrated load (kips) P = concentrated load nearest left reaction (kips) 1 P = concentrated load nearest right reaction, and of different magnitude than P 2 1 (kips) R = end beam reaction for any condition of symmetrical loading (kips) R1 = left end beam reaction (kips) R2 = right end orintermediate beam reaction (kips) R3 = right end beam reaction (kips) V = maximum vertical shear for any condition of symmetrical loading (kips) V = maximum vertical shear in left section of beam (kips) 1 = vertical shear at right reaction point, or to left of intermediate reaction point V 2 of beam (kips) = vertical shear at right reaction point, or to right of intermediate reaction point V 3 ofbeam (kips) = vertical shear at distance x from end of beam (kips) V x W = total load on beam (kips) a = measured distance along beam (in.) b = measured distance along beam which may be greater or less than a (in.) l = total length of beam between reaction points (in.) w = uniformly distributed load per unit of length (kips per in.) w1 = uniformly distributed load per unit of length nearest leftreaction (kips per in.) w2 = uniformly distributed load per unit of length nearest right reaction, and of different magnitude than w1 (kips per in.) x = any distance measured along beam from left reaction (in.) x1 = any distance measured along overhang section of beam from nearest reaction point (in.) ∆max = maximum deflection (in.) ∆a ∆x = deflection at point of load (in.) = deflection at any pointx distance from left reaction (in.)

∆x1 = deflection of overhang section of beam at any distance from nearest reaction point (in.)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION

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BEAM AND GIRDER DESIGN

BEAM DIAGRAMS AND FORMULAS Frequently Used Formulas The formulas given below are frequently required in structural designing. They are included herein for the convenience of thoseengineers who have infrequent use for such formulas and hence may find reference necessary. Variation from the standard nomenclature on page 4-187 is noted. BEAMS Flexural stress at extreme fiber: f = Mc / I = M / S Flexural stress at any fiber: f = My / I y = distance from neutral axis to fiber Average vertical shear (for maximum see below): v = V / A = V / dt (for beams and girders) Horizontalshearing stress at any section A-A: v = VQ / Ib Q = statical moment about the neutral axis of that portion of the cross section lying outside of section A-A b = width at section A-A (Intensity of vertical shear is equal to that of horizontal shear acting normal to it at the same point and both are usually a maximum at mid-height of beam.) Shear and deflection at any point: x and y are abscissa andordinate respectively of a point on the neutral d 2y EI 2 = M axis, referred to axes of rectangular coordinates through a selected dx point of support. (First integration gives slopes; second integration gives deflections. Constants of integration must be determined.) CONTINUOUS BEAMS (the theorem of three moments) Uniform load:  w1l3 w2l3   l1 l2  l1 l2 1 2  Ma + 2Mb  +  + Mc = − 1⁄4  + I1I2 I2   I1  I1 I2  Concentrated loads:  l1 l2  l1 l2 P1a1b1  a1  p2a2b2  b2  Ma + 2Mb  +  + Mc = − 1 +  − 1 +  I1 I2 I1  l1  I2  I2   I1 I2  Considering any two consecutive spans in any continuous structure: Ma, Mb, Mc = moments at left, center, and right supports respectively, of any pair of adjacent spans = length of left and right spans, respectively, of the pair l1 and...