Question

Problem 8 (Integration) For the beam and loading shown, use the double-integration method to determine (a) the equation of the elastic curve for segment AB of the beam, (b) the deflection midway between the two supports, (c) the slope at A, and (d) the slope at B. Assume that El is constant for the beam. - X A * 12*

Answer 1

(a) The equation of the elastic curve for segment AB,

[引學一需一

[$.? — x_7] -401 = 1

(b) The deflection midway between the two supports is,

PL Ymar = (0.0321) EI

(c) The slope at A is,

PL A = 12 EI

(d) The slope at B is,

0B = 6E1 -PL

The detailed solution is given below,

A P E£L2 t Figo Free body diagram {fy=0; RA+Ro = P - *z -KB -0 خمسمعك P (Like) - kgfelzo or = 36 Using equation ® Rhe P-3P Consider a portion of AB of long the from

we get M - - fx @ we know that or Eldy -- Intepate both sides, . EI dy = -f(x² + c d Indepsate above equation both sides, El Y = -f(x) + Gxtę Ely - - x + Qx tę applying boundary Conditions, Now I from equation o= o tot C₂ weget, » [eo] [atx= 4y=0]

Now, but above value in equation, 6 Now, but rake of G & C in equation (deflection REI Equation 6 & ® one the required equation of elastic Cure to 162 The deflection onidway between the two Ź Support is maximum asa.. at E Ymx is occurs, hence dy becomes . using equation , we get O= I I 1² x 2 T HEIL 3 I = 1m² or Xm = L = 0.5772 zen

but X-Im- 0.5774 in equation P L1(0.5771)-(0.5 221137 12EI P 3 [0.577 (0.577] REI E pL3 70.033127 EI ② slope at A; at A, x=0 using en 6 we get a 4E1 31 TOA = PL 12EL slepe at B Lat B, x = 11 using egr 0 we get لات