Slide 1

e.g. Determine the shape factor of the T-section.In this case the elastic and plastic N.A. Are not coincident therefore make a few assumptions.Let ye be the depth of the elastic N.A. from the top flange (passes through the centroid)Let yp be the depth of the plastic N.A. from the top flange

Step 1: find ye by taking moments about top surface of flangeStep 2: find moment of inertia I about elastic N.A.Step 3: Calculate the least value of Ze.

Step 4: Plastic N.A. divides cross-section into two equal areas therefore calculate flange and web areas to determine where yp will lie.Step 5: based on answer to Step 4, calculate yp. Step 6: calculate Zp.Step 7: calculate shape factor f.(Ans: f= 1.78)

Moment-curvature Relationships

Curvature of beam under elastic bending given by

At yield point

Combining the above equations gives (dimensionless)

Plot of the relationship (shown below) is linear until M=My .

As portions of beam become plastic plot becomes non-linear and flattens out as M approaches Mp.

It can be seen that as plastic region extends, small change in moment causes large increases in curvature

Within the range My M Mp a beam section has both fully plastic and elastic regions

In the plastic region yielding continues to happen without further increase in stress

In the elastic core stress continues to increase with deformationThis means that the elastic core controls deformation contained plastic flow

When M approaches Mp large deformations happen without increase in moment- unrestricted plastic flow

Plastic Hinges unrestricted plastic flow leads to formation of plastic hinges consider the beam below

maximum moment at midspan M=WL/4When the load is increased until M=Mp, Mp=WL/4Mid-span of the beam is fully plastic and there are regions of plasticity extending towards supports

Plasticity only occurs in those regions where moment greater than MyUnrestricted plastic flow now occurs at mid-spanBeam behaves as two rigid beams connected by a plastic hingeHinge allows beam portions to rotate relative to each other

Collapse load for the beam found as follows:

Mp=WL/4 , therefore

how do we find the length of the plastic region, Lp?