Home | | **Strength of Materials I** | | **Strength of Materials for Mechanical Engineers** | Spring Deflection and Wahl's factor

Spring striffness: The stiffness is defined as the load per unit deflection.
In order to take into account the effect of direct shear and change in coil curvature a stress factor is defined, which is known as Wahl's factor.

**SPRING DEFLECTION**

**Spring striffness: **The stiffness is defined as the load per unit deflection therefore

__Shear stress__

**WAHL'S FACTOR :**

In order to take into account the effect of direct shear and change in coil curvature a stress factor is defined, which is known as Wahl's factor.

K = Wahl' s factor and is defined as

Where C = spring index

= D/d

if we take into account the Wahl's factor than the formula for the shear stress becomes

**Strain Energy : **The strain energy is defined as the energy which is stored within a material** **when the work has been done on the material.

In the case of a spring the strain energy would be due to bending and the strain energy due to bending is given by the expansion

**Deflection of helical coil springs under axial loads**

Deflection of springs

**Example: **A close coiled helical spring is to carry a load of 5000N with a deflection of 50** **mm and a maximum shearing stress of 400 N/mm2 .if the number of active turns or active coils is 8.Estimate the following:

(i) wire diameter

(ii) mean coil diameter

(iii) weight of the spring.

Assume G = 83,000 N/mm2 ; = 7700 kg/m3

**solution :**

**(i) **for wire diametre if W is the axial load, then

Therefore,

D = .0314 x (13.317)3mm =74.15mm

D = 74.15 mm

__Weight__

Design of helical coil springs

**Helical spring design**

**Springs in Series: **If two springs of different stiffness are joined endon and carry a common** **load W, they are said to be connected in series and the combined stiffness and deflection are given by the following equation.

**Springs in parallel: **If the two spring are joined in such a way that they have a common** **deflection ‘x' ; then they are said t shared between the two springs and total load W = W1 + W2

stresses in helical coil springs under torsion loads

**Stresses under torsion**

**Shear Stress in the Shaft**

When a shaft is subjected to a torque or twisting, a shearing stress is produced in the shaft. The shear stress varies from zero in the axis to a maximum at the outside surface of the shaft.

The shear stress in a solid circular shaft in a given position can be expressed as:

*s = T r / Ip (1)*

*where*

*s = shear stress (MPa, psi)*

*T = twisting moment (Nmm, in lb)*

*r = distance from center to stressed surface in the given position (mm, in)*

*Ip = "polar moment of inertia" of cross section (mm4, in4)*

*The "polar moment of inertia" is a measure of an object's ability to resist torsion.*

**Circular Shaft and Maximum Moment**

Maximum moment in a circular shaft can be expressed as:

*Tmax = smax Ip / R (2)*

*Where*

Tmax = maximum twisting moment (Nmm, in lb)

smax = maximum shear stress (MPa, psi)

R = radius of shaft (mm, in)

Combining (2) and (3) for a solid shaft

Tmax = (p/16) smax D3 (2b)

Combining (2) and (3b) for a hollow shaft

Tmax = (p/16) smax (D4 - d4) / D (2c)

**Circular Shaft and Polar Moment of Inertia**

Polar moment of inertia of a circular solid shaft can be expressed as

*Ip = p R4/2 = p D4/32 (3)*

*D = shaft outside diameter (mm, in)*

Polar moment of inertia of a circular hollow shaft can be expressed as

*Ip = p (D4 - d4) /32* *(3b)*

*where*

*d = shaft inside diameter* *(mm, in)*

**Diameter of a Solid Shaft**

Diameter of a solid shaft can calculated by the formula

*D = 1.72 (Tmax/smax)1/3 (4)*

**Torsional Deflection of Shaft**

The angular deflection of a torsion shaft can be expressed as

*? = L T / Ip G (5)*

where

? = angular shaft deflection (radians) L = length of shaft (mm, in)

G = modulus of rigidity (Mpa, psi)

The angular deflection of a torsion solid shaft can be expressed as

? = 32 L T / (G p D4) (5a)

The angular deflection of a torsion hollow shaft can be expressed as

? = 32 L T / (G p (D4- d4)) (5b)

The angle in degrees can be achieved by multiplying the angle ? in radians with 180/p Solid shaft (p replaced)

?degrees ˜584 L T / (G D4) (6a)

Hollow shaft (p replaced)

?degrees ˜584 L T / (G (D4- d4) (6b)

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