Truss Analysis 5Db166

1. **Problem Statement:** We have a truss with supports at points $A$ and $E$. The truss is loaded with vertical point loads at joints $F$, $G$, $H$, and $I$. The vertical spacing between chords is $3$ m. The last 4 digits of the student number are $0409$, so we use these digits to determine the loads and parameters. Given: - $P_1 = 1A0 = 140$ N (since $A=4$) - $P_2 = 2B0 = 290$ N (since $B=9$) - $P_3 = 1C0 = 100$ N (since $C=0$ replaced by 1, so $1C0=110$ N) - $P_4 = 1D0 = 90$ N (since $D=9$) - Cross-sectional area of members = $2,AD0$ mm$^2$ = $2,4,0$ mm$^2$ = $240$ mm$^2$ - Thickness = $24$ mm - Diameter of rivets = $16$ mm - Allowable bearing stress = $15A,B0$ MPa = $15490$ MPa (using digits $A=4$, $B=9$) - Allowable shear stress = $7C.D0$ MPa = $7 0.90$ MPa = $7.90$ MPa 2. **Step 1: Calculate Reactions at Supports $A$ and $E$** - Use equilibrium equations: - Sum of vertical forces $\sum F_y = 0$ - Sum of moments about $A$ $\sum M_A = 0$ - Let reaction forces be $R_A$ and $R_E$ at supports $A$ and $E$ respectively. - Assume distances between joints horizontally are equal and known or given (assumed $3$ m between verticals). - Calculate total load: $$P_{total} = P_1 + P_2 + P_3 + P_4 = 140 + 290 + 110 + 90 = 630 \text{ N}$$ - Taking moments about $A$ (assuming distances for loads at $F$, $G$, $H$, $I$ as $3$, $6$, $9$, $12$ m respectively): $$\sum M_A = 0 = R_E \times 15 - (140 \times 3) - (290 \times 6) - (110 \times 9) - (90 \times 12)$$ - Calculate moment terms: $$140 \times 3 = 420$$ $$290 \times 6 = 1740$$ $$110 \times 9 = 990$$ $$90 \times 12 = 1080$$ - Sum moments: $$R_E \times 15 = 420 + 1740 + 990 + 1080 = 4230$$ - Solve for $R_E$: $$R_E = \frac{4230}{15} = 282 \text{ N}$$ - Use vertical force equilibrium: $$R_A + R_E = 630$$ $$R_A = 630 - 282 = 348 \text{ N}$$ 3. **Step 2: Calculate Axial Forces in Members $GD$, $GH$, and $CD$** - Use Method of Joints or Method of Sections. - Cross-sectional area $A = 240$ mm$^2 = 240 \times 10^{-6}$ m$^2$ - Calculate axial force $F$ in each member by resolving forces at joints. - For example, at joint $G$, sum forces in $x$ and $y$ directions to find forces in members $GD$ and $GH$. - Assume geometry and angles known or calculated from truss dimensions. - Calculate axial stress $\sigma$ in each member: $$\sigma = \frac{F}{A}$$ 4. **Step 3: Calculate Number of Rivets Required** - Diameter of rivet $d = 16$ mm - Area of one rivet cross-section: $$A_r = \frac{\pi}{4} d^2 = \frac{\pi}{4} (16)^2 = 201.06 \text{ mm}^2$$ - Bearing stress allowable $\sigma_b = 15490$ MPa - Shear stress allowable $\tau = 7.90$ MPa - Calculate rivet shear area (double shear assumed): $$A_{shear} = 2 \times A_r = 402.12 \text{ mm}^2$$ - Calculate rivet bearing area: $$A_{bearing} = d \times t = 16 \times 24 = 384 \text{ mm}^2$$ - Calculate rivet shear force capacity: $$F_{shear} = \tau \times A_{shear} = 7.90 \times 402.12 = 3176.75 \text{ N}$$ - Calculate rivet bearing force capacity: $$F_{bearing} = \sigma_b \times A_{bearing} = 15490 \times 384 = 5,949,760 \text{ N}$$ - Number of rivets required based on shear: $$n_{shear} = \frac{F_{member}}{F_{shear}}$$ - Number of rivets required based on bearing: $$n_{bearing} = \frac{F_{member}}{F_{bearing}}$$ - Use the larger $n$ for safety. 5. **Summary Table:** | Member | Axial Force $F$ (N) | Axial Stress $\sigma$ (MPa) | Rivets (Shear) | Rivets (Bearing) | Rivets Required | |--------|---------------------|-----------------------------|----------------|------------------|-----------------| | GD | $F_{GD}$ | $\sigma_{GD}$ | $n_{shear}$ | $n_{bearing}$ | $\max(n_{shear}, n_{bearing})$ | | GH | $F_{GH}$ | $\sigma_{GH}$ | $n_{shear}$ | $n_{bearing}$ | $\max(n_{shear}, n_{bearing})$ | | CD | $F_{CD}$ | $\sigma_{CD}$ | $n_{shear}$ | $n_{bearing}$ | $\max(n_{shear}, n_{bearing})$ | **Note:** Exact axial forces $F_{GD}$, $F_{GH}$, and $F_{CD}$ depend on truss geometry and load distribution, which must be calculated using the Method of Joints or Sections with the given truss diagram. **Final answers:** - Reactions: $R_A = 348$ N, $R_E = 282$ N - Axial stresses and rivet numbers depend on detailed force analysis.