Question: Q1 (40 marks): Draw the following beam’s shear and bending moment diagrams, then use the maximum shear to draw the shear stress distribution diagram for section A-A.
2 m
220 kN-m
Section A-A
220 mm
30 mm
220 mm
30 mm
top-left: beam diagram of a cantilever beam fixed at the left end with a downward clockwise end moment of 220 kN-m at the right end; section A-A is marked near the left side with a 2 m span, position_hint: top-left
top-right: T-section diagram labeled Section A-A with a top flange 220 mm wide and 30 mm thick, and a vertical web 30 mm wide and 220 mm deep, position_hint: top-right
1. **Problem Statement:**
We have a cantilever beam of length $2$ m fixed at the left end with a clockwise end moment of $220$ kN-m applied downward at the free right end. We need to draw the shear force and bending moment diagrams along the beam and then use the maximum shear force to find the shear stress distribution in the T-section at section A-A.
2. **Beam Loading and Reactions:**
- The beam is fixed at the left end (point A).
- A moment $M = 220$ kN-m is applied clockwise at the free end (point B).
- There are no vertical loads, so shear force $V$ along the beam is zero.
3. **Shear Force Diagram (SFD):**
- Since no vertical loads are applied, shear force $V(x) = 0$ everywhere along the beam.
4. **Bending Moment Diagram (BMD):**
- The bending moment at the free end $x=2$ m is $M_B = -220$ kN-m (clockwise moment is negative by sign convention).
- At the fixed end $x=0$, the moment is reaction moment $M_A = +220$ kN-m to balance the applied moment.
- The bending moment varies linearly along the beam:
$$M(x) = M_A - 0 \cdot x = 220 - 0 = 220 \text{ kN-m}$$
But since the moment is applied only at the free end, the moment diagram is constant at $220$ kN-m along the beam.
5. **Maximum Shear Force:**
- Maximum shear force $V_{max} = 0$ kN (no vertical loads).
6. **Shear Stress Distribution in Section A-A:**
- Since $V_{max} = 0$, the shear stress due to shear force is zero everywhere.
- However, bending moment causes normal stresses but not shear stresses.
- The problem asks to draw shear stress distribution using maximum shear, which is zero, so shear stress distribution is zero.
7. **Summary:**
- Shear force diagram is zero along the beam.
- Bending moment diagram is constant at $220$ kN-m along the beam.
- Shear stress distribution in section A-A is zero.
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**Note:** If the problem intended a different loading or if the moment is applied at the fixed end, please clarify. Based on given data, this is the solution.
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