A truss structure is loaded as shown in the figure below. Among the options given, which member in the truss is a zero-force member?

Given: F = 1000N

The correct option is BD.

Step-by-Step Explanation:

To identify the zero-force member in the given truss structure, we can analyze the equilibrium of the joints using the standard rules for zero-force members:

Rule for Zero-Force Members at a 3-Member Joint:
If exactly three members meet at a joint where two of the members are collinear, and there is no external load or support reaction applied at that joint, then the non-collinear (third) member must be a zero-force member.

Let us analyze Joint B based on the details shown in the figure:
1. Connected Members: Joint B connects three members: member BC, member BA, and member BD.
2. Collinearity: Members BC and BA are collinear, lying along the single continuous straight line CA (inclined at 45° to the horizontal member CD, with a total length of $\sqrt{2}\text{m}$).
3. External Forces & Supports: There are no external loads or supports located at Joint B. The external force F = 1000 N is applied at Joint A, and the support reactions are situated at Joint C (hinge support) and Joint D (roller support).
4. Angle: Member BD is perpendicular to the collinear line CA, as indicated by the 90° symbol at Joint B.

Now, let's verify this mathematically by setting up a local coordinate system at Joint B:
Let the x'-axis be aligned along the collinear members BC and BA.
Let the y'-axis be perpendicular to CA, pointing along the member BD.
Applying the equilibrium equation for Joint B along the perpendicular direction (y'-axis):

$$\sum F_{y\prime }=0$$

Since members BC and BA lie entirely along the x'-axis, they have no components acting along the y'-axis. Therefore, the only force contributing to this equilibrium equation is the internal force in member BD ($F_{\mathrm{BD}}$):

$$F_{\mathrm{BD}}=0$$

Thus, member BD is a zero-force member.