Chapter 17, Torsion of beams Video Solutions, Introduction to Aircraft Structural Analysis

Problem 1

A uniform, thin-walled, cantilever beam of closed rectangular cross-section has the dimensions shown in Fig. P.17.1. The shear modulus $G$ of the top and bottom covers of the beam is $18,000 \mathrm{~N} / \mathrm{mm}^2$ while that of the vertical webs is $26,000 \mathrm{~N} / \mathrm{mm}^2$. The beam is subjected to a uniformly distributed torque of $20 \mathrm{~N} \mathrm{~m} / \mathrm{mm}$ along its length. Calculate the maximum shear stress according to the Bred-Batho theory of torsion. Calculate also, and sketch, the distribution of twist along the length of the cantilever, assuming that axial constraint effects are negligible.

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Problem 2

A single-cell, thin-walled beam with the double trapezoidal cross-section shown in Fig. P.17.2, is subjected to a constant torque $T=90,500 \mathrm{~N} \mathrm{~m}$ and is constrained to twist about an axis through the point R. Assuming that the shear stresses are distributed according to the Bredt-Batho theory of torsion, calculate the distribution of warping around the cross-section. Illustrate your answer clearly by means of a sketch and insert the principal values of the warping displacements. The shear modulus $G=27,500 \mathrm{~N} / \mathrm{mm}^2$ and is constant throughout.

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Problem 3

A uniform thin-walled beam is circular in cross-section and has a constant thickness of $2.5 \mathrm{~mm}$. The beam is $2,000 \mathrm{~mm}$ long, carrying end torques of $450 \mathrm{~N} \mathrm{~m}$ and, in the same sense, a distributed torque loading of $1.0 \mathrm{~N} \mathrm{~m} / \mathrm{mm}$. The loads are reacted by equal couples $R$ at sections $500 \mathrm{~mm}$ distant from each end (Fig. P.17.3). Calculate the maximum shear stress in the beam and sketch the distribution of twist along its length. Take $G=30,000 \mathrm{~N} / \mathrm{mm}^2$ and neglect axial constraint effects.

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Problem 4

The thin-walled box section beam ABCD shown in Fig. P.17.4 is attached at each end to supports that allow rotation of the ends of the beam in the longitudinal vertical plane of symmetry but prevent rotation of the ends in vertical planes perpendicular to the longitudinal axis of the beam. The beam is subjected to a uniform torque loading of $20 \mathrm{~N} \mathrm{~m} / \mathrm{mm}$ over the portion BC of its span. Calculate the maximum shear stress in the cross-section of the beam and the distribution of angle of twist along its length, $G=70,000 \mathrm{~N} / \mathrm{mm}^2$.

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Problem 5

Figure P.17.5 shows a thin-walled cantilever box beam having a constant width of $50 \mathrm{~mm}$ and a depth which decreases linearly from $200 \mathrm{~mm}$ at the built-in end to $150 \mathrm{~mm}$ at the free end. If the beam is subjected to a torque of $1 \mathrm{kN} \mathrm{m}$ at its free end, plot the angle of twist of the beam at $500 \mathrm{~mm}$ intervals along its length and determine the maximum shear stress in the beam section. Take $G=25,000 \mathrm{~N} / \mathrm{mm}^2$.

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Problem 6

A uniform closed section beam, of the thin-walled section shown in Fig. P.17.6, is subjected to a twisting couple of $4,500 \mathrm{~N} \mathrm{~m}$. The beam is constrained to twist about a longitudinal axis through the center $\mathrm{C}$ of the semi-circular arc 12 . For the curved wall 12 , the thickness is $2 \mathrm{~mm}$ and the shear modulus is $22,000 \mathrm{~N} / \mathrm{mm}^2$. For the plane walls 23,34 , and 41 , the corresponding figures are $1.6 \mathrm{~mm}$ and $27,500 \mathrm{~N} / \mathrm{mm}^2$. (Note: $G t=$ constant.) Calculate the rate of twist in radians per millimetre. Give a sketch illustrating the distribution of warping displacement in the cross-section and quote values at points 1 and 4.

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Problem 7

A uniform beam with the doubly symmetrical cross-section shown in Fig. P.17.7, has horizontal and vertical walls made of different materials, which have shear moduli $G_a$ and $G_b$, respectively. If, for any material, the ratio mass density/shear modulus is constant, find the ratio of the wall thicknesses $t_a$ and $t_b$, so that, for a given torsional stiffness and given dimensions $a, b$, the beam has minimum weight per unit span. Assume the Bredt-Batho theory of torsion is valid. If this thickness requirement is satisfied find the $a / b$ ratio (previously regarded as fixed), which gives minimum weight for given torsional stiffness.

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Problem 8

The cold-formed section shown in Fig. P.17.8 is subjected to a torque of $50 \mathrm{Nm}$. Calculate the maximum shear stress in the section and its rate of twist. $G=25,000 \mathrm{~N} / \mathrm{mm}^2$.

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Problem 9

Determine the rate of twist per unit torque of the beam section shown in Fig. P.16.11 if the shear modulus $G$ is $25,000 \mathrm{~N} / \mathrm{mm}^2$. (Note that the shear center position is calculated in P.16.11.)

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Problem 10

Figure P.17.10 shows the cross-section of a thin-walled beam in the form of a channel with lipped flanges. The lips are of constant thickness $1.27 \mathrm{~mm}$, while the flanges increase linearly in thickness from $1.27 \mathrm{~mm}$ where they meet the lips to $2.54 \mathrm{~mm}$ at their junctions with the web. The web has a constant thickness of $2.54 \mathrm{~mm}$. The shear modulus $G$ is $26,700 \mathrm{~N} / \mathrm{mm}^2$ throughout. The beam has an enforced axis of twist $\mathrm{RR}^{\prime}$ and is supported in such a way that warping occurs freely but is zero at the mid-point of the web. If the beam carries a torque of $100 \mathrm{Nm}$, calculate the maximum shear stress according to the St. Venant theory of torsion for thin-walled sections. Ignore any effects of stress concentration at the corners. Find also the distribution of warping along the middle line of the section, illustrating your results by means of a sketch.

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Problem 11

The thin-walled section shown in Fig. P.17.11 is symmetrical about the $x$ axis. The thickness $t_0$ of the center web 34 is constant, while the thickness of the other walls varies linearly from $t_0$ at points 3 and 4 to zero at the open ends $1,6,7$, and 8 . Determine the St. Venant torsion constant $J$ for the section and also the maximum value of the shear stress due to a torque $T$. If the section is constrained to twist about an axis through the origin $\mathrm{O}$, plot the relative warping displacements of the section per unit rate of twist.

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Problem 11

If the length of the vertical web of the section shown in Fig. P.17.11 is labeled $L_{\text {web, }}$, use MATLAB to repeat the calculation of $J$ and $\tau_{\max }$ in Problem P.17.11 for values of $L_{\text {web }}$ ranging from $a$ to $4 a$ in increments of $a / 2$ (i.e., $a, 3 a / 2,2 a, \ldots, 4 a$ ).

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Problem 12

The thin-walled section shown in Fig. P.17.12 is constrained to twist about an axis through R, the center of the semi-circular wall 34 . Calculate the maximum shear stress in the section per unit torque and the warping distribution per unit rate of twist. Also compare the value of warping displacement at the point 1 with that corresponding to the section being constrained to twist about an axis through the point $\mathrm{O}$ and state what effect this movement has on the maximum shear stress and the torsional stiffness of the section.

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Problem 13

Determine the maximum shear stress in the beam section shown in Fig. P.17.13, stating clearly the point at which it occurs. Determine also the rate of twist of the beam section if the shear modulus $G$ is $25,000 \mathrm{~N} / \mathrm{mm}^2$.

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