Problem 1, 19pts

An initially straight beam is bent into a circle with radius R as shown in the figure. Material

fibers that are perpendicular to the axis of the undeformed beam are assumed to remain

perpendicular to the axis after deformation, and the beam's thickness and the length of its

axis are assumed to be unchanged. Under these conditions the deformation can be described

2₁ = (R-X₂) sin(¹), 2₂-R- (R-X2₂) cos(

(a) Calculate the deformation gradient field in the beam, expressing your answer as a

function of X₁, X₂,, and as components in the basis e₁,e₂, e shown.

(b) Calculate the Green Lagrange E* strain field in the beam.

(c) Calculate the infinitesimal strain field E in the beam.

(d) Compare the values of Lagrange strain and infinitesimal strain for two points that lie at

(X₁ = 0, X₂=h) and (X₁ = L, X₂-0). Explain briefly the physical origin of the difference

between the two strain measures at each point. Recommend maximum allowable values of

h/R and L/R for use of the infinitesimal strain measure in modeling beam deflections.

(e) Calculate the deformed length of an infinitesimal material fiber that has length lo and

orientation e, in the undeformed beam. Express your answer as a function of X₂.