**Strain Energy in Indeterminate Structures**

In indeterminate structures, the redundants are the unknown displacements or rotations that need to be solved to determine the equilibrium of the structure. The strain energy is a measure of the internal energy stored within a structure due to the deformation caused by external loads. The partial derivative of strain energy with respect to any of the redundants provides valuable information about the behavior of the structure.

**1. The partial derivative of strain energy with respect to any of the redundants is zero.**

This statement is not true. If the partial derivative of strain energy with respect to any of the redundants were zero, it would imply that there is no effect on the strain energy when the redundant is varied. However, this is not the case. The strain energy depends on the displacements or rotations of the redundants, and changing these values will result in a change in strain energy.

**2. The partial derivative of strain energy with respect to any of the redundants will give deflection in the direction of redundant.**

This statement is also not true. The partial derivative of strain energy with respect to a redundant provides information about the sensitivity of the strain energy to changes in that redundant. It does not directly give the deflection in the direction of the redundant. The deflection is determined by solving the equations of equilibrium and compatibility.

**3. The partial derivative of strain energy with respect to any of the redundants will give slope in the direction of redundant.**

This statement is more accurate. The partial derivative of strain energy with respect to a redundant gives the slope of the strain energy curve with respect to that redundant. This slope represents the change in strain energy per unit change in the value of the redundant. Therefore, it provides information about the sensitivity of the strain energy to changes in the redundant.

**4. The partial derivative of strain energy with respect to any of the redundants is a maximum.**

This statement is not true. The partial derivative of strain energy with respect to a redundant can be positive, negative, or zero. It depends on the specific configuration of the structure and the values of the redundants. The maximum or minimum value of the partial derivative does not necessarily indicate the maximum or minimum strain energy. The strain energy can have local maximum or minimum values at specific configurations, but these do not correspond to the maximum or minimum values of the partial derivative.

In summary, the correct statement is that the partial derivative of strain energy with respect to any of the redundants provides information about the sensitivity of the strain energy to changes in that redundant. It does not directly give the deflection or slope in the direction of the redundant, and it is not necessarily a maximum or minimum value.