Truss Analysis | Method of Joints | Method of Sections
Truss Analysis का उद्देश्य ट्रस के प्रत्येक सदस्य (member) में लगने वाले Force (Tension या Compression) को ज्ञात करना होता है। इसके लिए दो प्रमुख विधियाँ प्रयोग की जाती हैं:
- Method of Joints (संधि विधि)
- Method of Sections (खंड विधि)
1. Method of Joints (संधि विधि)
सिद्धांत
इस विधि में ट्रस के प्रत्येक Joint का अलग-अलग विश्लेषण किया जाता है।
प्रत्येक Joint पर निम्न Equilibrium Conditions लागू होती हैं:
- ΣFx = 0 (क्षैतिज बलों का योग = 0)
- ΣFy = 0 (ऊर्ध्वाधर बलों का योग = 0)
प्रक्रिया
Step 1: पूरे ट्रस के Support Reactions निकालें।
Step 2: ऐसे Joint से शुरुआत करें जहाँ अधिकतम दो Unknown Forces हों।
Step 3: Joint का Free Body Diagram (FBD) बनाएं।
Step 4: ΣFx = 0 तथा ΣFy = 0 लगाकर Member Forces निकालें।
Step 5: अगले Joint पर जाएँ और यही प्रक्रिया दोहराएँ।
Force की पहचान
- Positive Answer → Tension
- Negative Answer → Compression
लाभ
- सभी Members की Force आसानी से मिल जाती है।
- छोटे Truss के लिए सबसे उपयुक्त।
हानि
- बड़े Truss में समय अधिक लगता है।
2. Method of Sections (खंड विधि)
सिद्धांत
यदि केवल कुछ Members की Force चाहिए, तो पूरे Truss का विश्लेषण करने की आवश्यकता नहीं होती।
Truss को एक काल्पनिक Section से काटकर विश्लेषण किया जाता है।
Equilibrium Equations
- ΣFx = 0
- ΣFy = 0
- ΣM = 0
Moment Equation का प्रयोग सबसे अधिक किया जाता है।
प्रक्रिया
Step 1: Support Reactions निकालें।
Step 2: ऐसी Section लें जो अधिकतम तीन Unknown Members को काटे।
Step 3: किसी एक भाग का Free Body Diagram बनाएं।
Step 4: ΣM = 0 लगाकर पहले एक Force निकालें।
Step 5: ΣFx = 0 तथा ΣFy = 0 से बाकी Forces निकालें।
लाभ
- केवल आवश्यक Members की Force जल्दी मिल जाती है।
- बड़े Truss के लिए बहुत उपयोगी।
हानि
- सभी Members की Force निकालने के लिए उपयुक्त नहीं।
Method of Joints और Method of Sections में अंतर
| आधार | Method of Joints | Method of Sections |
|---|---|---|
| विश्लेषण | Joint-by-Joint | Section काटकर |
| Equations | ΣFx, ΣFy | ΣFx, ΣFy, ΣM |
| उपयोग | सभी Members की Force | कुछ Members की Force |
| गति | धीमी | तेज |
| उपयुक्त | छोटे Truss | बड़े Truss |
Tension और Compression
- Tension: Member खिंचता है।
- Compression: Member दबता है।
Whether you are planning a residential structural layout or a large-scale industrial roof, truss analysis boils down to figuring out the internal axial forces—tension and compression—traveling through each structural member. Because trusses are modeled with pin connections, we assume the members carry no bending moments, only axial loads.
To solve for these forces, structural engineers rely on two primary approaches.
Method of Joints
Think of this method as tearing the truss down and analyzing the equilibrium of forces at each individual pin (node).
The Mechanics: Because all the members meeting at a joint intersect at a single point, the forces are concurrent. This means there is no rotational moment to calculate. You isolate the joint as a particle and rely entirely on two equations of static equilibrium:
$$\sum F_x=0$$$$\sum F_y=0$$The Golden Rule: You can only start your analysis at a joint that has a maximum of two unknown forces and at least one known external load or support reaction.
When to Use It: When you need a complete structural profile and have to calculate the forces in every single member of the truss.
Method of Sections
Instead of hopping from joint to joint across the structure, this method allows you to jump straight to the middle of a truss by making an "imaginary slice" right through it.
The Mechanics: Slicing the truss divides it into two completely separate Free Body Diagrams (FBDs). Because the cut exposes forces that do not all intersect at a single point (non-concurrent), you gain access to the moment equation. You use three equations to solve the section:
$$\sum F_x=0$$$$\sum F_y=0$$$$\sum M=0$$The Golden Rule: Your imaginary cut should pass through no more than three members with unknown forces, as you only have three equations to solve for them.
When to Use It: When you need to quickly check the forces in a few specific members (like checking the stress on a critical mid-span diagonal) without wasting time calculating the entire perimeter first.
Direct Comparison
| Feature | Method of Joints | Method of Sections |
| Primary Use | Finding forces in the entire truss | Finding forces in targeted members |
| Equations | Two per joint ($\sum F_x=0$, $\sum F_y=0$) | Three per section ($\sum F_x=0$, $\sum F_y=0$, $\sum M=0$) |
| Force Type | Concurrent (intersect at the pin) | Non-concurrent (spread across the cut) |
| Speed | Slow and methodical | Fast for isolated checks |
| Error Risk | High (mistakes cascade) | Low (isolated to the section) |
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