Question:

The beam is made of an elastic-plastic material for which ${ \sigma }_{ Y }$ = 200 MPa. If the largest moment in the beam occurs within the center section a-a, determine the magnitude of each force P that causes this moment to be (a) the largest elastic moment and (b) the largest plastic moment.

Step-by-step

M = 2P

a) Elastic moment

$I = { \frac { 1 }{ 2 } } (0.1)(0.2^{3}) = 66.667(10^{-6}) m^{4}$

${ \sigma }_{ Y }=\frac { { M }_{ Y }c }{ I }$

${ M }_{ Y }=\frac { 200\left( { 10 }^{ 6 } \right) \left( 66.667 \right) \left( { 10 }^{ -6 } \right) }{ 0.1 } = 133.33 kN \cdot m$

From Eq. (1)

133.33 = 2 P

P = 66.7 kN

b) Plastic moment

${ { M }_{ p }=\frac { { bh }^{ 2 } }{ 4 } { \sigma }_{ Y } }$

${ =\frac { 0.1\left( { 0.2 }^{ 2 } \right) }{ 4 } \left( 200 \right) \left( { 10 }^{ 6 } \right) }$

= 200 kN $\cdot$ m

From Eq. (1)

200 = 2 P

P = 100 kN