Question

The options show frames consisting of rigid bars connected by pin joints. Which one of the frames is non-rigid? 

 

Hint:

The quickest way to solve such problems is by visual inspection. Look for any closed loops with four or more sides that are not divided into triangles by diagonal members. Such a loop indicates a non-rigid structure. In this case, the central rectangle in (C) is the immediate giveaway.

Answer 1

Step 1: Understanding the Concept:
A rigid frame (or perfect truss) is a structure that is stable and will not collapse under load. A non-rigid frame (or mechanism) can change its shape without any of its members changing length. For a pin-jointed plane frame to be rigid, it must satisfy a specific relationship between the number of members (\(m\)) and the number of joints (\(j\)). The simplest rigid shape is a triangle. Any frame made entirely of interconnected triangles is rigid.
Step 2: Key Formula or Approach:
The condition for a statically determinate and stable plane truss is: \[ m = 2j - 3 \] where \(m\) is the number of members and \(j\) is the number of joints.
- If \(m<2j - 3\), the frame is deficient or non-rigid (it's a mechanism).
- If \(m = 2j - 3\), the frame is a perfect or statically determinate frame (rigid).
- If \(m>2j - 3\), the frame is redundant or statically indeterminate (also rigid, but with extra members).
Alternatively, we can visually inspect the frames. A frame is non-rigid if it contains any sub-structure that is a polygon with more than three sides that is not internally braced (triangulated).
Step 3: Detailed Explanation:
Let's analyze each option:
- (A) This frame is composed entirely of triangles. It is a rigid structure. Let's check with the formula: \(j=5\), \(m=7\). Then \(2j-3 = 2(5)-3 = 7\). Since \(m = 2j-3\), it is a perfect frame.
- (B) This frame is also composed entirely of triangles. It is a rigid structure. Let's check with the formula: \(j=6\), \(m=9\). Then \(2j-3 = 2(6)-3 = 9\). Since \(m = 2j-3\), it is a perfect frame.
- (C) This frame has three bays. The left and right bays are triangulated and rigid. However, the central bay is a rectangle made of four pin-jointed bars. A rectangle is not a rigid shape; it can easily deform into a parallelogram. This makes the entire structure non-rigid. Let's check with the formula: \(j=8\), \(m=11\). Then \(2j-3 = 2(8)-3 = 13\). Here, \(m<2j-3\) (\(11<13\)), confirming the frame is deficient and non-rigid.
- (D) This frame is composed entirely of triangles. It is a rigid structure. Let's check with the formula: \(j=6\), \(m=9\). Then \(2j-3 = 2(6)-3 = 9\). Since \(m = 2j-3\), it is a perfect frame.
Step 4: Final Answer:
Frame (C) is the non-rigid frame.
Step 5: Why This is Correct:
The presence of the un-braced rectangular section in frame (C) makes it a mechanism, capable of changing shape under load without any member deforming. The other three frames are composed entirely of triangular elements, which is the fundamental requirement for a rigid pin-jointed frame.