Deformation and Stress Fields in Pure Bending

Visualize the displacement, normal stress, and stored elastic energy produced by pure bending of an elastic bar

A rectangular elastic bar of length 2𝐿 and height 2ℎ is subjected to equal moments of magnitude 𝑀 applied at both ends, resulting in a constant bending moment along the bar (pure bending). The bar is assumed to behave according to linear elasticity and is modeled under plane stress conditions. Using an analytical displacement field, compute and visualize the resulting deformation and normal stress distribution in the bar.

The variables 𝐿 and ℎ represent half of the bar length and half of its height, respectively:

In[1]:=

{h,L}={1,4};

To provide a clear reference for the problem setup, the undeformed geometry of the bar and the applied bending moments are visualized. The rectangle represents the bar cross-section, while the curved arrows indicate equal moments applied at both ends:

In[2]:=

Graphics[{LightGray,Rectangle[{-L,-h},{L,h}],Black,Arrow@Table[{-L-0.5hSin[t],0.5hCos[t]},{t,0,π,0.1π}],Arrow@Table[{L+0.5hSin[t],0.5hCos[t]},{t,0,π,0.1π}],Text["M",{-L-h,0}],Text["M",{L+h,0}]},ImageSize500]

Out[2]=

The material behavior is characterized by Young's modulus and Poisson's ratio, which determine the stiffness and lateral contraction of the bar under deformation:

In[3]:=

{young,ν}={3010^6,0.25};

Specify the magnitude of the applied moment:

In[4]:=

M=1;

An analytical expression for the displacement field gives the horizontal and vertical displacements of each point in the bar as functions of the spatial coordinates:

In[5]:=

u=(3M/(4youngh^3)){2xy,L^2-x^2-νy^2}

Out[5]=



20000000

16-

-0.25

40000000



The deformed configuration of the bar is visualized by plotting the displacement field over the original domain. The resulting shape illustrates the curvature produced by pure bending under a constant moment:

In[6]:=

VectorDisplacementPlot[u,{x,-L,L},{y,-h,h},ImageSize500]

Out[6]=

The normal stress acting along the length of the bar (the 𝑥-direction) is computed next. For pure bending, this stress varies linearly with the vertical coordinate and changes sign across the mid-height of the bar:

In[7]:=

σxx=(3M)/(2h^3)y;

The elastic strain energy density is formed from the normal stress and integrated over the bar domain to obtain the total strain energy stored under the applied moment:

In[8]:=

energyDensity=σxx^2/(2young);U=Integrate[energyDensity,{x,-L,L},{y,-h,h}]//FullSimplify

Out[8]=

5000000

The stress distribution is visualized on the deformed configuration by coloring the displaced bar according to the normal stress acting along the length of the bar:

In[9]:=

VectorDisplacementPlot[{u,σxx},{x,-L,L},{y,-h,h},PlotLegendsAutomatic]

Out[9]=

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Deformation and Stress Fields in Pure Bending

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Deformation and Stress Fields in Pure Bending

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