Design of shafts against loadings, Lecture notes of Machine Design

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Prepared by: Dr. Gagandeep Bhardwaj, Assistant Professor, MED Email: gagandeep.med@thapar.edu Contact No. 8954388548

DESIGN OF SHAFTS

• The term ‘ transmission shaft ’ is usually refers to a rotating machine element, circular in cross-section, which supports transmission elements like gears, pulleys and sprockets, and transmits power.
• The shaft is always stepped with maximum diameter in the middle portion and minimum diameter at the two ends, where bearings are mounted.
• The steps on the shaft provide shoulders for positioning transmission elements like gears, pulleys and bearings.

DESIGN OF SHAFT

• Spindle: A spindle is a short rotating shaft. The term ‘spindle’ originates from the round tapering stick on a spinning wheel, on which the thread is twisted.

DESIGN OF SHAFT

• Countershaft: It is a secondary shaft, which is driven by the main shaft and from which the power is supplied to a machine component. Often, the countershaft is driven from the main shaft by means of a pair of spur or helical gears.

Line shaft: A line shaft consists of a number of shafts, which are connected in axial direction by means of couplings. Line shafts were popular in workshops using group drive. In group drive construction, a single electric motor drives the line shaft. A number of pulleys are mounted on the line shaft and power is transmitted to individual machines by different belts.

DESIGN OF SHAFT

STANDARD DIAMETERS OF SHAFT

DESIGN OF SHAFT

Transmission shafts are subjected to: axial tensile force, bending moment or torsional moment and their combinations. The design of transmission shaft consists of determining the correct shaft diameter from strength and rigidity considerations.

Design of Shaft on Strength Basis : When the shaft is subjected to axial tensile force, the tensile stress is given by,

When the shaft is subjected to pure bending moment, the bending stresses are given by,

DESIGN OF SHAFT

Case I: In this case, the shaft is subjected to a combination of axial force, bending moment and torsional moment.

Case II: In this case, the shaft is subjected to a combination bending moment and torsional moment without any axial force.

The principal stress σ 1 is given by,

The principal shear stress τ max is given by,

DESIGN OF SHAFT

The shaft can be designed on the basis of maximum principal stress theory or maximum shear stress theory. We will apply these theories to transmission shaft subjected to combined bending and torsional moments.

(i) Maximum Principal Stress Theory: The maximum principal stress is σ 1. Since the shaft is subjected to bending and torsional moments without any axial force,

The permissible value of maximum principal stress is given by,

The maximum principal stress theory gives good predictions for brittle materials. Shafts are made of ductile material like steel and therefore, this theory is not applicable to shaft design.

DESIGN OF SHAFT

Equivalent Torsional Moment:

The expression is called ‘ equivalent ’ torsional moment. The equivalent torsional moment is defined as the torsional moment, which when acting alone, will produce the same torsional shear stress in the shaft as under the combined action of bending moment ( Mb ) and torsional moment ( Mt ).

Equivalent Bending Moment:

The expression is called ‘ equivalent ’ bending moment. The equivalent bending moment is defined as the bending moment, which when acting alone, will produce the same bending stresses (tensile and compressive) in the shaft as under the combined action of bending moment ( Mb ) and torsional moment ( M t ).

DESIGN OF SHAFT

Design of Shaft on Torsional Rigidity Basis :

• The shafts are designed on the basis of either torsional rigidity or lateral rigidity.
• A transmission shaft is said to be rigid on the basis of torsional rigidity , if it does not twist too much under the action of an external torque.
• Similarly, the transmission shaft is said to be rigid on the basis of lateral rigidity , if it does not deflect too much under the action of external forces and bending moment. In certain applications, like machine tool spindles, it is necessary to design the shaft on the basis of torsional rigidity , i.e., on the basis of permissible angle of twist per metre length of shaft. The angle of twist θr (in radians) is given by,

ASME CODE FOR SHAFT DESIGN

Equivalent Torsional Moment

The expression is called ‘ equivalent ’ torsional moment when the shaft is subjected to fluctuating loads.

Equivalent Bending Moment

The expression is called ‘ equivalent ’ bending moment when the shaft is subjected to fluctuating loads.

NUMERICAL PROBELMS

Problem 1 : A solid circular shaft is subjected to a bending moment of 3000 N-m and a torque of 10000 N-m. The shaft is made of 45C8 steel having ultimate tensile stress of 700 MPa and a ultimate shear stress of 500 MPa. Assuming a factor of safety as 6, determine the diameter of the shaft.

Given: M = 3000 N-m = 3 × 106 N-mm; T = 10 000 N-m = 10 × 106 N-mm; σ tu = 700 MPa = 700 N/mm^2 ; τ u = 500 MPa = 500 N/mm^2

NUMERICAL PROBELMS

Neglecting the weight of the pulley, the downward force at the pulley B is (P 1 + P 2 ) or 3676.47 N.

Similarly, the force in the horizontal plane at the pulley C is (P 3 + P 4 ) or 7352.94 N.

The force and bending moment in vertical plane

NUMERICAL PROBELMS

The force and bending moment in horizontal plane

The resultant bending moment is given by,

The resultant bending moment diagram and torsional moment diagram are shown