Long and short spans of a two way slab are $${l_{\text{y}}}$$ and $${l_{\text{x}}}$$ and load on the slab acting on strips parallel to $${l_{\text{x}}}$$ and $${l_{\text{y}}}$$ be wx and wy respectively. According to Rankine Grashoff theory A. $$\frac{{{{\text{w}}_{\text{x}}}}}{{{{\text{w}}_{\text{y}}}}} = \frac{{{l_{\text{y}}}}}{{{l_{\text{x}}}}}$$ B. $$\frac{{{{\text{w}}_{\text{x}}}}}{{{{\text{w}}_{\text{y}}}}} = {\left( {\frac{{{l_{\text{y}}}}}{{{l_{\text{x}}}}}} \right)^2}$$ C. $$\frac{{{{\text{w}}_{\text{x}}}}}{{{{\text{w}}_{\text{y}}}}} = {\left( {\frac{{{l_{\text{y}}}}}{{{l_{\text{x}}}}}} \right)^4}$$ D. None of these

The correct answer is: $\frac{{{{\text{w}}{\text{x}}}}}{{{{\text{w}}{\text{y}}}}} = {\left( {\frac{{{l_{\text{y}}}}}{{{l_{\text{x}}}}}} \right)^2}$

Rankine Grashoff theory states that the ratio of the loads on the two spans of a two-way slab is equal to the square of the ratio of the spans. This is because the bending moment in a two-way slab is proportional to the square of the load. Therefore, the load on the longer span must be greater than the load on the shorter span in order to produce the same bending moment in both spans.

The following is a brief explanation of each option:

• Option A: This option is incorrect because it states that the ratio of the loads is equal to the ratio of the spans. However, as explained above, the ratio of the loads is equal to the square of the ratio of the spans.
• Option B: This option is incorrect because it states that the ratio of the loads is equal to the square of the square of the ratio of the spans. This is not correct because the bending moment in a two-way slab is proportional to the square of the load, not the square of the square of the load.
• Option C: This option is incorrect because it states that the ratio of the loads is equal to the fourth power of the ratio of the spans. This is not correct because the bending moment in a two-way slab is proportional to the square of the load, not the fourth power of the load.
• Option D: This option is correct because it states that the ratio of the loads is equal to the square of the ratio of the spans. This is correct because the bending moment in a two-way slab is proportional to the square of the load.