The 2 kN horizontal force at the top of the 1 m post is replaced on the beam axis by what?

A 2 kN axial force and a 2 kN·m couple

What does the thrust (axial force) diagram show for the beam?

2 kN tension from A up to the post, then zero beyond

2 kN compression for the whole beam

The maximum negative moment is -4.00 kN·m at the roller. How is it obtained from the right side?

1.5 kN triangular load x 2 m plus the 1 kN·m end couple

Why is the maximum positive moment 2.94 kN·m at x = 2.5 m?

The shear force changes sign (from +0.38 to -4.62 kN) under the 5 kN load

2066 Chaitra (old course) · 16 marks

Beam thrust and bending moment diagrams with max moments

Short answer

Reactions: H_A=2 kN leftwards, V_A=0.38 kN upward, R_B=6.12 kN upward. Axial force (thrust) is 2 kN tension from A to the vertical post (0–1.5 m) and zero thereafter. Maximum sagging bending moment = 2.94 kN·m at 2.50 m from A; maximum hogging bending moment = –4.00 kN·m at the roller support (4 m).

Calculate maximum positive and maximum negative bending moment and draw thrust diagram and bending moment diagram for the beam as shown in the figure. [Figure: beam on a hinge support and a roller support; a vertical post of height 1 m carries a 2 kN horizontal force; a 5 kN downward load; a triangular load of 1 kN/m at the right end with a couple applied; dimensions 1.5 m, 1 m, 1.5 m, 1.5 m, 1.5 m.]

Assumptions (taken from the figure):

Beam from x = 0 to 7 m: hinge A at x = 0 and roller B at x = 4 m. A 1 m post at x = 1.5 m carries a 2 kN horizontal force (→) at its top; this is equivalent to a 2 kN axial force on the beam plus a clockwise couple of 2 × 1 = 2 kN·m.
5 kN downward at x = 2.5 m; triangular load from 0 at x = 4 m to 1 kN/m at x = 7 m (total 1.5 kN, acting at x = 6 m); clockwise couple of 1 kN·m at the free end x = 7 m.

Sign convention: N is positive in tension; V is positive when the forces on the part to the left of the section act upward; M is quoted with the tension face named (sagging = bottom, hogging = top).

Support reactions

$\Sigma F_x = 0$ : $H_A$ = 2.00 kN ←
$\Sigma M_A = 0$ : $4R_B = 2 + 5(2.5) + 1.5(6) + 1 = 24.5$ , so $R_B$ = 6.12 kN ↑ (6.125)
$\Sigma F_y = 0$ : $V_A = 6.5 - 6.125$ = 0.38 kN ↑ (0.375)

Axial force, shear force and bending moment at the key sections

Section	N (kN)	V (kN)	M (kN·m)
At A (x = 0)	2.00	0.38	0.00
Just left of post (x = 1.5)	2.00	0.38	0.56 (bottom)
Just right of post	0.00	0.38	2.56 (bottom)
Just left of 5 kN (x = 2.5)	0.00	0.38	2.94 (bottom)
Just right of 5 kN	0.00	-4.62	2.94 (bottom)
Just left of roller B (x = 4)	0.00	-4.62	-4.00 (top)
Just right of roller B	0.00	1.50	-4.00 (top)
At x = 5.5 m	0.00	1.12	-1.94 (top)
Free end, left of couple (x = 7)	0.00	0.00	-1.00 (top)

Thrust diagram

N = +2 kN (tension) from A to the post (x = 0 to 1.5 m) and zero from the post to the free end.

Shape of the diagrams

Between loads V is constant; under a UDL it varies linearly and under a triangular load parabolically. M is one degree higher than V (linear, parabolic, cubic respectively); V jumps at point loads and reactions, and M jumps at couples (by 2 kN·m at the post).
AFD: constant between loads on a member, changing only where a load component acts along the member.
BMD is drawn on the tension side of each member and is zero at the hinge support A.

Salient features

Maximum positive (sagging) M = 2.94 kN·m at x = 2.50 m
Maximum negative (hogging) M = -4.00 kN·m at x = 4.00 m (over the roller B)
Point of contraflexure (M = 0) at x = 3.14 m from A

Sketches (sagging +ve above the line)

SFD:




                              *****************
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                   |          |
                   |          |
                   |          |
                   ***********|

BMD:


           ********
           |       **
          *|         **
**********-------------*----------------------------
                        **                  ********
                          *           ******
                           **   ******
                             ***

✓ Answer checked by Claude Opus 5.5 (high effort)

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