Question

Using Macaulay's method, calculate the deflection under the concentrated load and the slope at C of a propped cantilever beam loaded as shown in the Figure 2 below in terms of El. 80 kN 50 KN 30 kN/m A D 2 m 4 m 4 m B С

Please help. Have started but am struggling

Answer 1

Method to solve weling Macaulay's Method, . Σ P 1 3a R Live) A + B a b VA X He и, ч or use Initially cut a section before the end of the beam either moving deft to right right to left sign in My equation which differentiate different terme closely and we get to knows, which term to be neglected, El dig - W (arb) My A duz I-terman I-term III-term 4. aca и y At point At point Q At point "R Neglect rect and & ord term arnaby only - No tem to be neglected and form Neglect good и ч ab

Sign s Convention an I to R My Mx IR tol tre Vx V In both causes. Vy n n

Applying boundary condition in C₂=0 6) (no term to be neglected) 103 6 @ point D x=0; deflection, y=0 → put. This in deflection equation (neglecting II 4 Y Herm av (X<4;8)) Constant, @ point A x= 10 m; deflection, yol put this in deflection equation 0 = Voi +10.6, +0 =30(e– 50.610-49 – 80.619-09 500. V6 + 10.4 330 a point A → delom; slope, dy=0 put this in slope equation neglected) Volvo + , - 30 (tele 50.410-4)2_ 80 (10-8)2 50, V6 + C, On solving wis & co equations Vo = 138.58 KN ---Lin 12,500 + 1800 + 320 con da (no term to be 0 = 2 2 = 5000 + 900 + 160 ung o We get; and c. - 869

of x=8m . in deflection (Ų Deflection under concentrated load 80 KN (@point B) put value of b, G, C2 & equation E.T.y 869+8+0 – 30118 indicates downwared deflection land] 138:58 8 24 Deflection, ly -779.84 ЕТ. -ve sign

of en slope equation : (2) Slope at point C :- put the value No , c, & x=4m (neglect III term) as x<8) (**)+(-869) – 30 EI. dy 138.58 10 (493) - 5oferty da Slope, dy -ve -80.36 Еу. sign Indicates slope in anti-clockwise direction dse of (3) Deflection under concentrated lood of 50 KN ( @ point () put the value Up, C, C2, &x=4m in deflection equation (neglect I term as x<8) $() +(-86944) +0 - 30 Deflection, y 2317.8133 sign Indicates downward deflection EILY = 138.58 sofern 24 EL

Answers:

(1) Deflection at point B is in downward direction which equals to;

$y=\frac{-779.84}{EI}$   

(2) Slope at point C is in Anti-Clockwise direction which equals to;

dy dr -80.36 ΕΙ

(3) Deflection at point C is in downward direction which equals to;

$y=\frac{-2317.8133}{EI}$

(Note - The unit of deflection is meters provided that "EI" should be in kN-m^​​​2​​)