VIPSolutionsWhich option correctIre ein emer lel Lalli ERR = = Fs ES EE: = = : = .. =e Olea Ane = _ _ = -— = a a EEE EEE Eee. 8. £ [i =f 00] == 2 Kl=2x10 Tbofin (£] oa 5 foo 11} fi 5 fr -10 0) EE) = = p £]=10 Ibfin CEE ee 7 00-11} 8 2 [1-10 0] 12-10 0 ’ = 1bht X]=10 Ibfin Ky 5 a 1 0 0 LL] — JLB Coven — 01:10:53 * g o [= Question 12 @ Max. score: 100. Calculate the global stiffness matrix for the 3-bar composition with the node count indicated in the figure. The first and second nodes are fixed. Each element has a length of 50 in. For elements 1. 2, and 3, the young modules and rea are given E =50x10' ps and 4 =21n B,=40x10" psand 4, =1in E, =100x10° ps and 4, =lin 0:00 4% » . > — < »< >< > Sin 0m Sin
Question:
Which option correct
Ire ein emer lel Lalli ERR = =
Fs ES
EE: =
= : =
.. =e
Olea Ane = _ _ = -— = a
a EEE EEE Eee. 8.
£ [i =f 00]
==
2 Kl=2x10 Tbofin
(£] oa
5 foo 11}
fi
5 fr -10 0)
EE)
= = p
£]=10 Ibfin
CEE ee
7 00-11}
8
2 [1-10 0]
12-10
0 ’ = 1bht
X]=10 Ibfin
Ky 5 a
1 0 0 LL]
—
JLB Coven — 01:10:53 * g o [=
Question 12 @ Max. score: 100.
Calculate the global stiffness matrix for the 3-bar composition with the node count indicated in the figure. The first and second nodes are fixed. Each element has a length
of 50 in. For elements 1. 2, and 3, the young modules and rea are given
E =50x10' ps and 4 =21n
B,=40x10" psand 4, =1in
E, =100x10° ps and 4, =lin
0:00 4%
» . > —
< »< >< >
Sin 0m Sin
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Asked by: Sanjay hipparakar
Created at: 2025-09-18 14:20:18
Asked by: Sanjay hipparakar
Created at: 2025-09-18 14:20:18
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## Question Analysis
You are asked to calculate the **global stiffness matrix** for the 3-bar system shown, with the following data:
- Each bar has a length \( L = 50 \) in.
- Element 1: \( E_1 = 50 \times 10^6 \) psi, \( A_1 = 2 \) in²
- Element 2: \( E_2 = 40 \times 10^6 \) psi, \( A_2 = 1 \) in²
- Element 3: \( E_3 = 100 \times 10^6 \) psi, \( A_3 = 1 \) in²
## Stiffness Calculation
The local stiffness for a bar is:
\[
k = \frac{EA}{L}
\]
Calculate for each:
- \( k_1 = \frac{50 \times 10^6 \times 2}{50} = 2 \times 10^6 \) lb/in
- \( k_2 = \frac{40 \times 10^6 \times 1}{50} = 0.8 \times 10^6 \) lb/in
- \( k_3 = \frac{100 \times 10^6 \times 1}{50} = 2 \times 10^6 \) lb/in
Now, the global stiffness matrix for 3 elements (4 nodes) in 1D bar:
\[
[K] = \begin{bmatrix}
k_1 & -k_1 & 0 & 0 \\
-k_1 & k_1 + k_2 & -k_2 & 0 \\
0 & -k_2 & k_2 + k_3 & -k_3 \\
0 & 0 & -k_3 & k_3 \\
\end{bmatrix}
\]
Substitute \( k_1 = 2 \times 10^6 \), \( k_2 = 0.8 \times 10^6 \), \( k_3 = 2 \times 10^6 \):
\[
[K] = 10^6 \begin{bmatrix}
2 & -2 & 0 & 0 \\
-2 & 2 + 0.8 & -0.8 & 0 \\
0 & -0.8 & 0.8 + 2 & -2 \\
0 & 0 & -2 & 2 \\
\end{bmatrix}
\]
\[
[K] = 10^6 \begin{bmatrix}
2 & -2 & 0 & 0 \\
-2 & 2.8 & -0.8 & 0 \\
0 & -0.8 & 2.8 & -2 \\
0 & 0 & -2 & 2 \\
\end{bmatrix}
\]
## Match With Options
The options shown in your image are all for \( [K] = 2 \times 10^6 \) or \( 10^6 \) and with integer entries (1, 2, -1, etc), which corresponds to a case where all \( k \)'s are equal, which is **not the case here**!
### None of the options exactly match the correct values found above.
#### But, if you must pick the one that matches the **form** (even if the numerical values are not exact):
- The **first option** (with \( 2 \times 10^6 \) and integer matrix) matches the **form** of the global stiffness matrix **if all \( k \) values were equal to \( 2 \times 10^6 \)**.
#### So, **Option 1** is the most structurally correct, but none are numerically correct for your given data.
---
## **Summary**
- **Correct form:** Option 1 (first option in your list)
- **Numerical accuracy:** None match exactly; the actual values should not be all 1’s and 2’s.
---
### **If you must select one for a multiple-choice test, choose Option 1 (the top option with \( 2 \times 10^6 \)).**
---
**If you want the exact correct matrix, refer to the calculation above.**
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