Mechanical Engineering Principles

6 MECHANICAL ENGINEERING PRINCIPLES

Now try the following exercise

Exercise 2 Further problems on strain

1. A wire of length 4.5 m has a percentage
   strain of 0.050% when loaded with a ten-
   sile force. Determine the extension in the
   wire. [2.25 mm]


2. A metal bar 2.5 m long extends by
   0.05 mm when a tensile load is applied to
   it. Determine (a) the strain, (b) the
   percentage strain.

[(a) 0.00002 (b) 0.002%]

3. An 80 cm long bar contracts axially
   by 0.2 mm when a compressive load is
   applied to it. Determine the strain and the
   percentage strain. [0.00025, 0.025%]


4. A pipe has an outside diameter of 20 mm,
   an inside diameter of 10 mm and length
   0.30 m and it supports a compressive
   load of 50 kN. The pipe shortens by
   0.6 mm when the load is applied.
   Determine (a) the compressive stress,
   (b) the compressive strain in the pipe
   when supporting this load.

[(a) 212.2 MPa (b) 0.002 or 0.20%]

5. When a circular hole of diameter 40 mm
   is punched out of a 1.5 mm thick metal
   plate, the shear stress needed to cause
   fracture is 100 MPa. Determine (a) the
   minimum force to be applied to the
   punch, and (b) the compressive stress in
   the punch at this value.

[(a) 18.85 kN (b) 15.0 MPa]

6. A rectangular block of plastic material
   400 mm long by 15 mm wide by 300 mm
   high has its lower face fixed to a bench
   and a force of 150 N is applied to the
   upper face and in line with it. The upper
   face moves 12 mm relative to the lower
   face. Determine (a) the shear stress, and
   (b) the shear strain in the upper face,
   assuming the deformation is uniform.

[(a) 25 kPa (b) 0.04% or 4%]

1.7 Elasticity, limit of proportionality

and elastic limit

Elasticityis the ability of a material to return to its

original shape and size on the removal of external

forces.

Plasticity is the property of a material of being

permanently deformed by a force without breaking.

Thus if a material does not return to the original

shape, it is said to be plastic.

Within certain load limits, mild steel, copper,

polythene and rubber are examples of elastic mate-

rials; lead and plasticine are examples of plastic

materials.

If a tensile force applied to a uniform bar of mild

steel is gradually increased and the corresponding

extension of the bar is measured, then provided the

applied force is not too large, a graph depicting these

results is likely to be as shown in Figure 1.7. Since

the graph is a straight line,extension is directly

proportional to the applied force.

Extensionx

Force

F

Figure 1.

The point on the graph where extension is no

longer proportional to the applied force is known as

thelimit of proportionality. Just beyond this point

the material can behave in a non-linear elastic man-

ner, until theelastic limitis reached. If the applied

force is large, it is found that the material becomes

plastic and no longer returns to its original length

when the force is removed. The material is then

said to have passed its elastic limit and the result-

ing graph of force/extension is no longer a straight

line. Stress,σ=F/A, from Section 1.5, and since,

for a particular bar, areaAcan be considered as a

constant, thenF∝σ.

Strainε =x/L, from Section 1.6, and since for

a particular barLis constant, thenx∝ε. Hence

for stress applied to a material below the limit of