Mechanics of materials

Group members:

Rana Muhammad jahanzaib ME 123079Usama bilal me123097Khalid munir hashmi me123078

Objectives for Today

By the end of the session, participants will have some understanding with: Longitudinal shear on a beam element of

arbitrary shape. (Topic 6.6) Plastic deformation. (Topic 6.8) Application. (Numerical problem # 6.37)

Longitudinal shear on a beam of arbitrary shape

Consider: What If?

In section 6.4:

We have examined the distribution of the vertical components txy on a transverse section of a beam and found to have constant value in the web of the beam (S-beam , W-beam).

But what about the horizontal component txz of the stresses ??

Consider the prismatic beam AB:

Plastic Deformations

• The section becomes fully plastic (yY = 0) at the wall when

pY MMPL 2

3

• For PL > MY , yield is initiated at B and B’. For an elastoplastic material, the half-thickness of the elastic core is found from

2

2

3

11

2

3

c

yMPx YY

moment elastic maximum YY c

IM

• Recall:

• Preceding discussion was based on normal stresses only

• Shear load is carried by the elastic core,

A

P

byAy

y

A

PY

Yxy

2

3

2 where12

3

max

2

2

• Consider horizontal shear force on an element within the plastic zone,

0 dAdAH YYDC

Therefore, the shear stress is zero in the plastic zone.

• As A’ decreases, tmax increases and may exceed tY

Numerical problem # 6.37

Your turn!

Any Question?“For Every Action there Is a Reaction, Which Is Equal In Intensity But Opposite In Direction”