Question

Problem 8 (10 points) A bar with two segments AC and CB each have a cross section of an equilateral triangular. The dimensions are as follows: Segment AC has sides $a_{AC}$ = 60 mm and length $L_{AC}$ = 2.5 m. Segment CB has sides $a_{CB}$ = 45 mm and length $L_{CB}$ = 1.5 m. The bar has an allowable shear strength of $\tau_{all}$ = 50 MPa and an allowable twist per unit length of $\theta_{all}$ = 1.5° per meter. The torsion relations are shown in the table below. (a) Determine the maximum permissible applied torque $T_{max}$ assuming G = 80 GPa (5 points) (b) Determine the corresponding angle of twist between the end sections, $\phi_{max}$, at the max torque value calculated in part (a) (5 points) T A 2.5 m Section AC C 1.5m 60 mm $\Delta$ 45 mm Section CB Cross section Maximum shearing stress Angle of twist per unit length 2b $\tau_A = \frac{2T}{\pi ab^2}$ $\theta = \frac{(a^2+b^2)T}{\pi a^3 b^3 G}$ A For circular bar: a=b $\tau_A = \frac{20T}{a^3}$ $\theta = \frac{46.2T}{a^4 G}$ Equilateral triangle $\tau_A = \frac{T}{\alpha ab^2}$ $\theta = \frac{T}{\beta ab^3 G}$ a/b $\beta$ $\alpha$ 1.0 0.141 0.208 1.5 0.196 0.231 2.0 0.229 0.246 2.5 0.249 0.256 3.0 0.263 0.267 4.0 0.281 0.282 5.0 0.291 0.292 10.0 0.312 0.312 $\infty$ 0.333 0.333 T

          Problem 8 (10 points)
A bar with two segments AC and CB each have a cross section of an equilateral triangular. The
dimensions are as follows: Segment AC has sides $a_{AC}$ = 60 mm and length $L_{AC}$ = 2.5 m. Segment CB
has sides $a_{CB}$ = 45 mm and length $L_{CB}$ = 1.5 m. The bar has an allowable shear strength of $\tau_{all}$ =
50 MPa and an allowable twist per unit length of $\theta_{all}$ = 1.5° per meter. The torsion relations are shown
in the table below.
(a) Determine the maximum permissible applied torque $T_{max}$ assuming G = 80 GPa (5 points)
(b) Determine the corresponding angle of twist between the end sections, $\phi_{max}$, at the max torque
value calculated in part (a) (5 points)
T
A
2.5 m
Section AC
C
1.5m
60 mm
$\Delta$
45 mm
Section CB
Cross section
Maximum
shearing stress
Angle of twist
per unit length
2b
$\tau_A = \frac{2T}{\pi ab^2}$
$\theta = \frac{(a^2+b^2)T}{\pi a^3 b^3 G}$
A
For circular bar: a=b
$\tau_A = \frac{20T}{a^3}$
$\theta = \frac{46.2T}{a^4 G}$
Equilateral triangle
$\tau_A = \frac{T}{\alpha ab^2}$
$\theta = \frac{T}{\beta ab^3 G}$
a/b
$\beta$
$\alpha$
1.0
0.141
0.208
1.5
0.196
0.231
2.0
0.229
0.246
2.5
0.249
0.256
3.0
0.263
0.267
4.0
0.281
0.282
5.0
0.291
0.292
10.0
0.312
0.312
$\infty$
0.333
0.333
T

problem 8 10 points a bar with two segments ac and cb each have a cross section of an equilateral triangular the dimensions are as follows segment ac has sides aac 60 mm and length lac 25 32012

Added by Ashley W.

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