3 plastic analysis of beams
DESCRIPTION
plastic analysis of beamsTRANSCRIPT
Effect of Axial Load on Plastic Moment
Plastic Analysis of BeamsFailure (Collapse) Mechanisms
the failure mechanism describes the state in which a beam acts as a mechanism at the point of failure in this state we have sections of the beam rotating with respect to each other about plastic hinges a statically determinate beam will fail completely when a single plastic hinge is formed but indeterminate beams may have to develop several before collapse occurs as shown below e.g. consider the indeterminate beam below
the elastic bending moment diagram
the moment at the support is larger so it reaches a value equal to the plastic moment Mp first and a plastic hinge is formed as load is further increased the beam now acts as if it is simply supported until plastic moment reached at B and another hinge is formed beam can take NO MORE load
the corresponding bending moment diagram is
moment distribution occurs in indeterminate beams i.e. when support A reaches Mp then moment stays constant (at plastic hinge) and bending increases at B instead
Remember: Yield condition bending moment at any point must not exceed plastic moment at that point
moment redistribution:- causes change in shape of bending moment diagram (elastic vs plastic)- increases ultimate strength of structure (as one section fails then other sections have to carry extra load)
ultimate load can be found using statics:
Considering BC
and, therefore
plastic analysis of indeterminate simpler than elastic analysis since we dont have to use elastic bending moment diagram at all we simply find positions of plastic hinges and calculate using statics
NB: -ultimate load not affected by imperfections and sinking supports - cannot use superposition- loads assumed to be applied simultaneously and ratio between them remains constant
Virtual Work Method
considering the propped cantilever again
at collapse the beam is in equilibrium with plastic hinge at A and B lets say that AB rotates by a small amount ; BC also rotates by the same amount
vertical displacement at B is L/2 (small angles) angle at B is 2 reactions at A and C do not displace vertically the only force doing work is WU internal forces doing work are MP at A and B (resist rotation)
and as before
since plastic hinges form at regions of peak bending moment it follows that for beams with a series of point loads, hinges will form under the loads
e.g. calculate the minimum value of w required to cause collapse
largest moments occur at A and some point C between A and B rotation of AB and BC are and respectively vertical deflection of C is
total load on AC is wx acting at x/2 and will be displaced a distance d/2 total load on CB is w(L-x) and its centroid also displaced d/2 using virtual work
NB: B is a hinge- rotates- no plastic hinge formation combining equations above
finally
equation yields more than one value of we need to find minimum value of w
which reduces to
solving this quadratic and ignoring the negative value of x gives
substituting value of x back into expression for w gives
now use lower bound theorem to check we have critical mechanism taking moments about Awhich gives
taking moments about B
vertical equilibrium is satisfied
now considering RHS and taking moments about C
substituting for RB and w gives
same result from considering LHS load satisfies vertical and moment equilibrium bending moment at any distance x1 from B
and
from which we get
substituting for RB, x1 and w into
givestherefore yield criterion is met
so the yield mechanism assumed is correct.