To determine the maximum concentrated load $P$ that the girder can support, we need to check the maximum bending and shear stresses produced by the loads and ensure they do not exceed the allowable bending and shear stresses. Here’s the approach:

1. Determine Moment Capacity:

   • Use the maximum allowable bending stress, $F_b$, and the section modulus $S$ to find the maximum moment $M$ the girder can withstand.

   $M_{\text{max}} = F_b \cdot S$

   where: $S = \frac{I_x}{\frac{d}{2}} = \frac{446 \times 10^6 \, \text{mm}^4}{\frac{465}{2} \, \text{mm}}$

   After calculating $S$, we’ll substitute it into the formula for $M_{\text{max}}$.

2. Shear Capacity:

   • Calculate the shear capacity using the allowable shear stress $F_v$.

   $V_{\text{max}} = F_v \cdot A$

   where $A$ is the cross-sectional area of the web, given as $A = b_f \cdot t_w$.

3. Moment and Shear Diagrams:

   • Given that the loads are at the third points, the maximum bending moment for each load configuration can be calculated at the center of the span.
   • Similarly, for shear, consider the maximum shear force just to the left or right of each load application point.

4. Determine the Maximum Load $P$:

   • Using the maximum allowable moment and shear calculated from steps 1 and 2, solve for the maximum load $P$ that the girder can support under these conditions.

I'll calculate each of these steps to find the maximum permissible value of $P$.The calculations yield the following results:

1. Maximum Bending Moment Capacity ($M_{\text{max}}$): $M_{\text{max}} = 314.6 \, \text{kN} \cdot \text{m}$

2. Maximum Shear Capacity ($V_{\text{max}}$): $V_{\text{max}} = 210.2 \, \text{kN}$

Using these capacities, we can now determine the allowable load $P$ by distributing the bending moment and shear according to the load arrangement (loads at third points) and checking for compliance with the bending and shear capacities. Let me proceed with this final calculation.The maximum concentrated load $P$ that the girder can support, given the bending and shear constraints, is approximately:

$P = 104.9 \, \text{kN}$

This load represents the maximum value each of the two loads $P$ can be, ensuring both the bending and shear stresses remain within allowable limits.

Would you like further details on any step, or help with related questions?

Related Questions

1. How would the allowable load change if the span length were increased?
2. What if the allowable bending stress $F_b$ increased by 10%—how would that affect $P$?
3. How can we adjust the girder dimensions to allow a higher $P$?
4. How would this calculation change if the loads were uniformly distributed instead of concentrated?
5. What is the significance of the modulus of elasticity $E$ in this problem?

Tip

When designing girders or beams, always verify both bending and shear capacities, as either can become the limiting factor depending on the load configuration.