A Seminalytical Approach to Large Deflections in Compliant Beams under Point Load

<table>The dimensionless normalized rotation angle of the beam versus <svg height="7.9499998" id="M76" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" id="x1D6FC"></path></g>
</svg> and <svg height="15.95" id="M77" style="vertical-align:-2.55255pt" version="1.1" viewbox="0 0 7.8249998 15.95" width="7.8249998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.713)"><path d="M430 689q0 -42 -40.5 -66t-86.5 -28q-79 -65 -137 -176t-58 -211q0 -52 21 -79t60 -41q25 -9 62 -18q77 -20 77 -89q0 -34 -22 -69.5t-42 -54.5t-50 -44l-22 23q60 63 60 111q0 45 -50 55q-42 8 -67 18q-55 20 -83.5 56t-28.5 105q0 113 68 224t164 194q-52 1 -86 23.5
t-34 51.5q0 20 21 40t53 30l17 -12q-13 -17 -13 -36t19 -38.5t58 -22.5q85 65 123 65q17 0 17 -11z" id="x1D701"></path></g>
</svg> with respect to a reference value of 0.644930, for <svg height="7.9499998" id="M78" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" id="x1D6FC"></path></g>
</svg> from 0 to 1.5, and <svg height="15.95" id="M79" style="vertical-align:-2.55255pt" version="1.1" viewbox="0 0 7.8249998 15.95" width="7.8249998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.713)"><path d="M430 689q0 -42 -40.5 -66t-86.5 -28q-79 -65 -137 -176t-58 -211q0 -52 21 -79t60 -41q25 -9 62 -18q77 -20 77 -89q0 -34 -22 -69.5t-42 -54.5t-50 -44l-22 23q60 63 60 111q0 45 -50 55q-42 8 -67 18q-55 20 -83.5 56t-28.5 105q0 113 68 224t164 194q-52 1 -86 23.5
t-34 51.5q0 20 21 40t53 30l17 -12q-13 -17 -13 -36t19 -38.5t58 -22.5q85 65 123 65q17 0 17 -11z" id="x1D701"></path></g>
</svg> from 0 to1.</table>

Mathematical Problems in Engineering

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Figure 3

Figure 3: A Seminalytical Approach to Large Deflections in Compliant Beams under Point Load 