Built-in Beams 151 Taking moments about A, Mg+4Rg-(40×1.6)-(30×2.4×2.8)-(-42.12)=0 Mg=-(67.9×4)+64+201.6-42.12=-48.12kNm i.e.again in the opposite direction to that assumed in Fig.6.8. (Alternatively,and more conveniently,this value could have been obtained by substitution into the original Macaulay expression with x =4,which is,in effect,taking moments about B. The need to take additional moments about A is then overcome.) Substituting into the Macaulay deflection expression, 0-21皆+9 EI +6-3x-1.6]3-[x-1.6] Thus,under the 40kN load,where x =1.6(and neglecting negative Macaulay terms), y= 10[-421×2.56+44.1×4-0-0 E 2 6 23.75×103 14×106=-1.7×10-3m =-1.7mm The negative sign as usual indicates a deflection downwards. Example 6.3 Determine the fixing moment at the left-hand end of the beam shown in Fig.6.9 when the load varies linearly from 30kN/m to 60 kN/m along the span of 4m. 30 kN/m w kN/m 60 kN/m -dx -(L-x) 4m- fig.6.9. Solution From§6.4 MA= w'x(L-x)2 dx Now 103=(30+7.5x)103N/mBuilt-in Beams 151 Taking moments about A, MB+4RB-(40X 1.6)-(30~2.4~2.8)-(-42.12)=0 .. i.e. again in the opposite direction to that assumed in Fig. 6.8. (Alternatively, and more conveniently, this value could have been obtained by substitution into the original Macaulay expression with x = 4, which is, in effect, taking moments about B. The need to take additional moments about A is then overcome.) MB = - (67.9 x 4) + 64 + 201.6 - 42.12 = - 48.12 kN m Substituting into the Macaulay deflection expression, xz 44.12 20 GYy 2 6 3 = -42.1- + ~ - - [X - 1.613 - $[x - 1.614 El Thus, under the 40 kN load, where x = 1.6 (and neglecting negative Macaulay terms), - (42.1 x 2.56) (44.1 x 4.1) y =E[ EZ 2 + 6 -0-01 23.75 x lo3 14 x lo6 =- = -1.7~ 10+m = - 1.7mm The negative sign as usual indicates a deflection downwards. Example 6.3 Determine the fixing moment at the left-hand end of the beam shown in Fig. 6.9 when the load varies linearly from 30 kN/m to 60 kN/m along the span of 4 m. 30 kNhI kN/rn A Fig. 6.9. Solution From $6.4 Now w' = (30 + F) lo3 = (30 + 7.5x)103 N/m