Strength of Materials - Lecture Notes / Mehmet Zor

1

23 Agust 2024

4.3

TORSION OF BARS WITH RECTANGULAR OR

THIN PROFILE CROSS-SECTION�

tvid- 4.3

Strength of Materials - Lecture Notes / Mehmet Zor

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23 Agust 2024

Planar sections of non-circular shafts do not remain planar and the stress and strain distributions do not change linearly.

Maximum Shear Stress:

Unit Twist Angle :

Total Twist Angle:

Since the torsion of bars having rectangular cross-section is different and more complex than that of circular cross-section shafts, their solutions are made with the Theory of Elasticity or membrane analogy.

The obtained solution results are given below, depending on the section side ratios.

As seen in Figure 4.3.4, maximum stresses occur on the long side, minimum stresses occur on the short side, and the stresses become "zero" at the corners.

a: long side, b: short side

(occurs in the middle of the long side and at the outermost)

Shear flow distribution

4.3.1 Torsion of Rectangular Section Bars

Torsion of Bars with Rectangular or Profile Cross-Section

Polar Moment of Inertia :

Figure 4.3.2

Figure 4.3.3

Figure 4.3.4

Stress distribution in the section

(4.3.1)

(4.3.2)

(4.3.3)

(4.3.4)

Figure 4.3.1

Strength of Materials - Lecture Notes / Mehmet Zor

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23 Agust 2024

a = 50 mm, b = 20 mm, L = 1.2 m

Safe

a)

b)

Torsion of Bars with Rectangular or Profile Cross-Section

Solution:

From table 4.3.1 :

From equ. 4.3.1, polar moment of inertia:

From equ. 4.3.4, maximum shear stress:

From equ. 4.3.3, Total angle of twist:

Figure 4.3.5

Strength of Materials - Lecture Notes / Mehmet Zor

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23 Agust 2024

4.3.2 Torsion of Thin Profile Section Bars

The polar moment of inertia is the sum of the moments of inertia of each rectangle:

Unit Angle of twist of the Whole Bar :

Total Angle of twist of the Whole Bar :

Maximum shear stress in cross-section occurs at the outer part of the thinnest rectangle:

In this type of bars, the cross section is a combination of multiple thin rectangles.

The equations in Table 4.3.1 and 4.3.1-4.3.4, which are valid for a single rectangular section in article 4.3.1, are used for each rectangle forming the profile section.

The formulas are valid for all thin profiles, symmetrical or non-symmetrical.

Torsion of Bars with Rectangular or Profile Cross-Section

T

T

(4.3.5)

(4.3.4)

(4.3.2)

(4.3.3)

Figure 4.3.6

(a)

(b)

Figure 4.3.7

(a)

(b)

(c)

Strength of Materials - Lecture Notes / Mehmet Zor

5

23 Agust 2024

L=1.25m

For each rectangle: a / b =12. In table 4.3.1.. c₁ = c₂ = ⅓ =0.333 =c_1-1 = c_1-2 =c_2-1=c_2-2 for all values ​​of a/b greater than 10

Solution:

The cross-section consists of two different (horizontal and vertical) rectangles. Long side of each rectangle a=h=60mm, short side: b=t=5mm

(unsafe)

a)

b)

Torsion of Bars with Rectangular or Profile Cross-Section

(From equ. 4.3.4):

(From equ. 4.3.3):

;

t=5 mm, h=60 mm

Figure 4.3.8

(a)

(b)