Question

Consider the beam ABC of length L [m] in Figure 1 below, with simple supports at both ends. The beam supports a concentrated load P [N] at point B. You may assume the beam to be weightless in your analysis.

Figure 1: Schematic of beam ABC.

Part (a)

Determine the vertical reaction forces at points A and C in terms of P.

Part (b)

Determine expressions (in terms of P and L) for the shear force, V(x) and the bending moment, M(x) along the length of the beam. Here, x is defined as the distance [m] along the beam from point A.

Part (c)

Determine expressions (in terms of P, L, E, and I) for the deflection, v(x) and slope, v'(x) along the length of the beam.

Hint: The deflection is continuous at point B (its value is the same whether we are considering segment AB or segment BC). The same holds for the slope.

Answer 1

SOLUTION schematic of the beam l-a * . RA a>vertical reaction forces RAARB=P 0 EMA_0%-l RB + Pa RB = Pole from equation o P RA+Pole RA=P 2. Da P (l-a) C Pa by Expression for shear force & Bending moment Shear force at section x-x Ven=RA-P P (8) -? PC-%) =o- Pap ✓ Vana-Po/a //

20 Max - RAXOL - P ( x-a) =P(129) * -P(1-0) *(1-%) -P(x-a) = p - Pro The apa Po (1.1%) mm men c ) dy Ei dre? day dre? ('- ~ ) } { Po ( 12 2 13 ) } du? hel در) pa ę ? Ônoço aut naponta Integrating dy Pa yo no.qon boz x= 1 ter? slope 2 ) 13 Integrating the equation @

3 y h 13 2 م2 za + Case + C2 Pa Equation El deflection Exact values of constant ai & cz can be known numeric-information was given regarding .l, an & P. il