Understanding Moving Loads and Bending Moment

When loads move across a structural element like a girder, the internal forces, such as the bending moment, change at different points along the structure. For safe design, engineers need to determine the position of the moving load that causes the largest possible bending moment at any given section of the girder. This is known as finding the condition for the maximum bending moment at a section.

Maximum Bending Moment with a Moving Uniformly Distributed Load (UDL)

The question specifically addresses a scenario involving a uniformly distributed load (UDL) that is shorter than the girder's span. This UDL moves across the girder from left to right. The principle for achieving the maximum bending moment at a specific section differs based on the type of moving load.

For a point load, the maximum bending moment at a section typically occurs when the point load is directly positioned over that section.

However, for a UDL shorter than the span, the condition is different. The maximum bending moment at a particular section occurs when the UDL is positioned optimally relative to that section. The governing principle is as follows:

• Consider a section at a distance '$x$' from the left support and '$L-x$' from the right support, where '$L$' is the total span of the girder.
• This section divides the span into two segments with lengths '$x$' and '$L-x$'. The ratio of these segments is '$x : (L-x)$'.
• The maximum bending moment at this section occurs when the UDL is positioned such that the section divides the *load* in the same ratio as it divides the *span*. Since the UDL has a uniform intensity '$w$', this translates to the lengths of the UDL lying on either side of the section being in the same ratio as the span segments.

Mathematically, this condition can be expressed as:

$$ \frac{\text{Length of UDL to the left of the section}}{\text{Length of UDL to the right of the section}} = \frac{x}{L-x} $$

Evaluating the Given Options

Let's examine why the other options provided are not the correct condition for maximum bending moment for a UDL shorter than the span:

• Option 1: The head of the load reaches the section
  This condition is generally associated with achieving maximum shear force at a section, not necessarily the maximum bending moment, especially for distributed loads.
• Option 2: The tail of the load reaches the section.
  Similar to the first option, this position does not guarantee the maximum bending moment for a moving UDL.
• Option 3: The load position should be such that the section divides it equally on both sides.
  This implies the ratio of load on either side is 1:1. This specific scenario would only yield the maximum bending moment if the section was at the midpoint of the span and the UDL was positioned symmetrically around it. It's not the general condition.
• Option 4: The load position should be such that the section divides the load in the same ratio as it divides the span
  This statement accurately reflects the principle derived from influence line theory or equilibrium analysis for maximizing bending moments under moving UDLs shorter than the span. It ensures the load distribution relative to the section optimally contributes to the bending moment.

Therefore, the correct positioning of the UDL ensures that the ratio of the load segments on either side of the section matches the ratio of the span segments.