1. The problem: Find the ultimate moment capacity of an irregular beam. 2. Step 1: Understand that the ultimate moment capacity ($M_u$) of a beam is the maximum moment that the beam can resist before failure. 3. Step 2: To calculate $M_u$, we need the beam's cross-sectional shape, material properties, and reinforcement details (if any). The irregular shape means we must find the section's plastic moment capacity using integration or piecewise calculation of stresses. 4. Step 3: Calculate the cross-sectional properties such as the section modulus or moment of inertia, depending on available information. 5. Step 4: Apply the material's yield strength ($f_y$ for steel, $f_c'$ for concrete, etc.) to find the maximum stress. 6. Step 5: For irregular shapes, divide the cross section into simpler shapes, calculate individual moments about a neutral axis, sum those to get the resultant moment capacity. 7. Step 6: Use $$M_u = f_y \times Z_p$$ where $Z_p$ is the plastic section modulus of the irregular beam. 8. If reinforcement exists, include the reinforcement contribution by: $$M_u = A_s f_y (d - a/2)$$ where $A_s$ is the reinforcement area, $d$ is the effective depth, and $a$ is the depth of the equivalent stress block. 9. Conclude by combining all contributions according to the beam design code used. Note: Without specific beam geometry and properties, only the method can be outlined.