Deflections¶

CalcpadCE worksheets in this section deal with beam deflection under classical Euler-Bernoulli theory, where the curvature \(\kappa\) is proportional to the bending moment through the bending stiffness \(EI\). Each sheet integrates the curvature twice along the span to give the rotation \(\varphi(x)\) and deflection \(d(x)\) closed forms for a single member and load case.

A cantilever with a tip force and a cantilever under uniform load cover the two basic clamped-free responses, recovering \(d_{max} = F l^3 / (3 E I)\) and \(d_{max} = q l^4 / (8 E I)\) at the free end respectively. A simply supported beam with a mid-span point load and a simply supported beam under uniform load provide the pinned-pinned counterparts, with the central \(5 q l^4 / (384 E I)\) deflection used in serviceability checks.

Span, load magnitude, second moment of area \(I\), modulus \(E\) and a probe coordinate \(x\) are exposed as inputs, so \(d(x)\) and the maximum rotation \(\varphi_{max}\) can be read for any geometry without editing the formulas.

Cantilever with Concentrated Load¶

Rotation \(\varphi(x)\) and deflection \(d(x)\) along a cantilever loaded by a tip force \(F\), recovering the textbook tip values \(\varphi_{max} = F l^2 / (2 E I)\) and \(d_{max} = F l^3 / (3 E I)\).

Code:

'<div style = "max-width: 180mm;">
'<img style="width:165pt;" class="side" src="../../Images/mechanics/beams/deflection-cantilever-force.png" alt="deflection-cantilever-force.png">
'Beam length - 'l = ?'m
'Load - 'F = ?'kN
'Second moment of area - 'I = ?'cm<sup>4</sup>
'Modulus of elasticity - 'E = ?'GPa
'Functions of rotation and deflection along the beam
EI = E*I*0.01
φ(x) = F/EI*(l*x - x^2/2)
d(x) = F/(2*EI)*(l*x^2 - x^3/3)
#hide
PlotWidth = 600
PlotHeight = 150
#show
$Plot{-d(x)*10^3 @ x = 0 : l}
'Maximum deflection
d_max = d(l)*10^3'mm
'Maximum rotation
φ_max = φ(l)*10^3'rad·10³
'Calculate deflection at'x = ?'m
d(x)*10^3'mm
'</div>3    50  160000  30  3

Cantilever with Unifrom Load¶

Cantilever under a full-span uniform load \(q\), integrating the curvature to obtain \(\varphi(x)\) and \(d(x)\) with the tip deflection \(d_{max} = q l^4 / (8 E I)\).

Code:

'<div style = "max-width: 180mm;">
'<img style="width:165pt;" class="side" src="../../Images/mechanics/beams/deflection-cantilever-distributed-load.png" alt="deflection-cantilever-distributed-load.png">
'Beam length - 'l = ?'m
'Load - 'q = ?'kN/m
'Second moment of area - 'I = ?'cm<sup>4</sup>
'Modulus of elasticity - 'E = ?'GPa
'Functions of rotation and deflection along the beam
EI = E*I*0.01
φ(x) = q/(2*EI)*(x^3/3 - l*x^2 + l^2*x)
d(x) = q/(2*EI)*(x^4/12 - l*x^3/3 + l^2*x^2/2)
#hide
PlotWidth = 600
PlotHeight = 150
#show
$Plot{-d(x)*10^3 @ x = 0 : l}
'Maximum deflection
d_max = d(l)*10^3'mm
'Maximum rotation
φ_max = φ(l)*10^3'rad·10³
'Calculate deflection at'x = ?'m
d(x)*10^3'mm
'</div>3    30  160000  30  2

Simply Supported Beam with Concentrated Load¶

Simply supported beam with a mid-span point force \(F\), giving the maximum deflection \(d_{max} = F l^3 / (48 E I)\) and the support rotation \(\varphi_{max} = F l^2 / (16 E I)\).

Code:

'<div style = "max-width: 180mm;">
'<img style="width:165pt;" class="side" src="../../Images/mechanics/beams/deflection-simply-supported-beam-force.png" alt="deflection-simply-supported-beam-force.png">
'Beam length - 'l = ?'m
'Load - 'F = ?'kN
'Second moment of area - 'I = ?'cm<sup>4</sup>
'Modulus of elasticity - 'E = ?'GPa
'Functions of rotation and deflection along the beam
EI = 0.01*E*I
d(x) = F/(4*EI)*((l^2/4*x - x^3/3)*(x ≤ l/2) + (l^2/4*(l - x) - (l - x)^3/3)*(x > l/2))
φ(x) = F/(4*EI)*(l^2/4 - x^2)
#hide
PlotWidth = 600
PlotHeight = 150
#show
$Plot{-d(x)*10^3 @ x = 0 : l}
'Maximum deflection
d_max = d(l/2)*10^3'mm
'Maximum rotation
φ_max = φ(0)*10^3'rad·10³
'Calculate deflection at'x = ?'m
d(x)*10^3'mm
'</div>6    100 160000  30  3

Simply Supported Beam with Unifrom Load¶

Simply supported beam under a full-span uniform load \(q\), recovering the serviceability-relevant mid-span deflection \(d_{max} = 5 q l^4 / (384 E I)\) and the support rotation \(\varphi_{max} = q l^3 / (24 E I)\).

Code:

'<div style = "max-width: 180mm;">
'<img style="width:165pt;" class="side" src="../../Images/mechanics/beams/deflection-simply-supported-beam-distributed-load.png" alt="deflection-simply-supported-beam-distributed-load.png">
'Beam length - 'l = ?'m
'Load - 'q = ?'kN/m
'Second moment of area - 'I = ?'cm<sup>4</sup>
'Modulus of elasticity - 'E = ?'GPa
'Functions of rotation and deflection along the beam
EI = E*I*0.01
φ(x) = q/EI*(x^3/6 - l*x^2/4 + l^3/24)
d(x) = q/EI*(x^4/24 - l*x^3/12 + l^3/24*x)
#hide
PlotWidth = 600
PlotHeight = 150
#show
$Plot{-d(x)*10^3 @ x = 0 : l}
'Maximum deflection
d_max = d(l/2)*10^3'mm
'Maximum rotation
φ_max = φ(0)*10^3'rad·10³
'Calculate deflection at'x = ?'m
d(x)*10^3'mm
'</div>6    30  160000  30  3

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