Question

Problem 3. (25 points) For the cantilever beam shown below (A is a fixed reaction), take El as constant. You may assume the beam remains elastic. "x" is referenced from A moving right. 4 k/ft 12 ft (a) (4 pts). Calculate the reactions at the fixed connect (point A) (b) (4 pts). What are the Boundary Conditions for displacement (y and 0) for this beam? Where do they occur? (c) (8 pts). Develop the equation for the bending moment as a function of x (i.e. M(x)) using the Euler-Bernoulli Beam Theory (integration method). Start with w(x) = x/3 and integrate till you find M(x).

Answer 1

Solution posted below.

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YRIL Yix) HA a) Cantilever beam with fixed support at A. 12 ft » рее Тух Efy=0 RA-( x 4 x 12) = 0 = RA: 24K GEMAO - (2 x4 x 12 ) ( 22 x 12) - MA= 0 MA= – 192 kaft G E frio НА - С b) Boundary condition for displacement. are those condition which define the Calue of displacements Chor ) cet a particular pt. in ccro at maximum.

for cantilever beam. defliction, y = 0 Cat =o) Y = Tmax Cat x=L) slope - 0 Cat &=L) moment, m=0 cat u=2) Shear terce, U=0 Cat &=L) C) Using Euler- Bernoulli beam theory we know El deco) 2: wman Turn Lead ( ₂ x 4x12) = 24k ULT = 4.x .WCW) Part mar s by similar triangle lan uk 12 9. K Сүч Integrating above can wort & Es de MC @ Again integrate above can wort.de EI d'w = x + ( x + (2 0 F ax2 18

Integrate ean X k2 Cox+cs X Er din + - cx Integrate above can ③ EI w C.x3 C2 x², xs & 360 Can & Cy C. - 2 =) Using boundary condro at x=0 =) y=0= w w fer l is cusce deplection at x=0, 0 = 0 {o= aw) de at K = L = ma duz x=L=1284, m=0 [ dem 1 O= 12 + C, 12 + C2 at x = L=12 ft, v-o [- EI dew =] oc, [C, --24) E at k = L = clu2

O= 123 + (-24x12) + C2 1 Using aan Er du eta + CX+C Ş "El desem 18 - Mur X – 24x + 192 marsson mcx) = – x3 24 x – 192 9 & 24.102 18