AFD, SFD, BMD of frame with 40 kN/m UDL

Reading of the figure: A is a hinge at (0, 0), B is at (3, 4) (the member AB is 5 m long, inclined; vertical rise 2.5 + 1.5 = 4 m), BC is horizontal from x = 3 to x = 7 m (4 m) and carries 40 kN/m. C–D–E is the vertical column at x = 7 m, with E a roller at the base (0 m) and D 2 m above E. Loads: 10 kN acting perpendicular to AB at 1.875 m from A (the point at height 1.5 m), and 60 kN horizontal, towards the left, at D.

The 10 kN has components 10(0.8, −0.6) = (8 kN →, 6 kN ↓), since AB makes 53.13° with the horizontal.

Support reactions

ΣM_A = 0:  R_E × 7 + 60 × 2 − 160 × 5 − 6 × 1.125 − 8 × 1.5 = 0      (160 = 40 × 4 acts at x = 5)
           7 R_E = 800 + 6.75 + 12 − 120 = 698.75   ->  R_E = 99.82 kN ↑
ΣFx = 0:   A_x + 8 − 60 = 0                          ->  A_x = 52.00 kN →
ΣFy = 0:   A_y + 99.82 − 6 − 160 = 0                 ->  A_y = 66.18 kN ↑

Internal forces (tension positive for N; sagging positive for M)

Member AB (inclined, direction cosines 0.6, 0.8)

• Axial: N = −(52 × 0.6 + 66.18 × 0.8) = −84.14 kN (compression), constant, since the 10 kN is perpendicular to AB.
• Shear (perpendicular to AB): V = 1.89 kN from A to the 10 kN load, then 11.89 kN beyond it (both in the same sense).
• Moment: M_A = 0; at the load point (1.875 m from A): 1.89 × 1.875 = 3.55 kN·m; at B: 3.55 + 11.89 × 3.125 = 40.71 kN·m (tension on the outer, upper-left face).

Member BC (horizontal, 4 m)

• Axial: N = −60.00 kN (compression), constant.
• Shear: V_B = +60.18 kN, V(x) = 60.18 − 40x, V_C = −99.82 kN. V = 0 at x = 1.50 m from B.
• Moment: M(x) = −40.71 + 60.18x − 20x²: M_B = −40.71, zero at x = 1.03 m and 1.98 m, maximum sagging +4.55 kN·m at x = 1.50 m, M_C = −120.00 kN·m (hogging, tension on top).

Member CE (vertical)

• Axial: N = −99.82 kN (compression) over the whole length.
• Shear: V = 0 from E to D; V = 60 kN from D to C.
• Moment: M = 0 from E to D; then increases linearly to 120 kN·m at C (tension on the right, outer face).

Diagram values

BMD (tension side)            AFD            SFD of BC
  B  40.71 -- 120 C       all compressive   +60.18
   \  (+4.55)   |         -84.14 AB          \
  3.55            |        -60.00 BC            \_ 0 at 1.5 m
   A          D (0)        -99.82 CE               \ -99.82

Salient features: M is maximum at C (120 kN·m, hogging); M is zero at A, at E, and along ED; the member EC carries no shear below the 60 kN load; the BMD is parabolic under the UDL, with a small sagging peak of 4.55 kN·m; the shear passes through zero where the sagging moment peaks; the joint moments are in equilibrium (at C, 120 kN·m in the column = 120 kN·m in the beam, at B 40.71 kN·m in both members).