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Propped Cantilevers

CalcpadCE worksheets in this section cover the propped cantilever — a beam fixed at one end and simply supported at the other. The extra support makes the system one degree statically indeterminate, so the fixed-end moment and the two reactions depend on the geometry and load alone (the bending stiffness cancels for a prismatic beam), and equilibrium by itself is not enough to find them.

Each sheet works through the force-method result for one load case and reports the fixed-end moment \(M_A\), the support reactions \(V_A\) and \(V_B\), and the location and magnitude of the in-span peak. Concentrated actions are treated as a point force at distance \(a\) from the fixed end and a point moment at the same position, with both formulas written in terms of the dimensionless ratio \(k = b / l\). The distributed cases include a uniform load over the full span, a linearly varying load between intensities \(q_1\) and \(q_2\), and a partial uniform load of length \(b\) offset from the supports.

Span, load magnitudes, distances and a probe coordinate \(x_1\) are exposed as inputs, so \(M(x_1)\) and \(V(x_1)\) can be read off the bending and shear diagrams at any cross-section.

Concentrated Force

Force-method solution for a propped cantilever carrying a point force \(F\) at distance \(a\) from the fixed support, with the in-span peak at the load position.

Code:
'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="propped-cantilever-beam-concentrated-force.png" class="side" src="../../Images/mechanics/beams/propped-cantilever-beam-concentrated-force.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'F = ?'kN
'Distances -'a = ?'m,
#post
b = l - a'm,'k = b/l
'<p><b>Internal forces</b></p>
'Bending
M_A = F*a*k*(1 + k)/2'kN·m
M_max = F*b*(2 - 3*k + k^3)/2'kN·m
'Shear
V_A = F*k/2*(3 - k^2)'kN
V_B = F - V_A'kN
'<p><b>Diagrams</b></p>
#hide
PlotWidth = 600
PlotHeight = 150
#show
'Calculate internal forces at'x_1 = ?'m
#pre

#post
'Bending
M(x) = -M_A + V_A*x - F*(x - a)*(x > a)
$Plot{-M(x) @ x = 0 : l}
M(x_1)'kN·m
'Shear
V(x) = V_A - F*(x > a)
$Plot{V(x) @ x = 0 : l}
V(x_1)'kN
#show
'</div>10   10  4   1
Rendered Output:
propped-cantilever-beam-concentrated-force.png

Input data

Beam Length - l = 10  m

Load - F = 10  kN

Distances - a = 4  m,

b = l − a = 10 − 4 = 6 m, k = bl = 610 = 0.6

Internal forces

Bending

MA = F · a · k ·  ( 1 + k ) 2 = 10 · 4 · 0.6 ·  ( 1 + 0.6 ) 2 = 19.2 kN·m

Mmax = F · b ·  ( 2 − 3 · k + k3 ) 2 = 10 · 6 ·  ( 2 − 3 · 0.6 + 0.63 ) 2 = 12.48 kN·m

Shear

VA = F · k2 ·  ( 3 − k2 )  = 10 · 0.62 ·  ( 3 − 0.62 )  = 7.92 kN

VB = F − VA = 10 − 7.92 = 2.08 kN

Diagrams

Calculate internal forces at x1 = 1  m

Bending

M ( x )  = -MA + VA · x − F ·  ( x − a )  ·  ( x > a ) 

Plot

M  ( x1 )  = M  ( 1 )  = -11.28 kN·m

Shear

V ( x )  = VA − F ·  ( x > a ) 

Plot

V  ( x1 )  = V  ( 1 )  = 7.92 kN

Concentrated Moment

Propped cantilever loaded by a concentrated moment \(M\) at distance \(a\), giving a constant shear and a bending diagram with a jump of \(M\) across the load section.

Code:
'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="propped-cantilever-beam-concentrated-moment.png" class="side" src="../../Images/mechanics/beams/propped-cantilever-beam-concentrated-moment.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'M = ?'kN·m
'Distances -'a = ?'m,
#post
b = l - a'm,'k = b/l
'<p><b>Internal forces</b></p>
'Bending at support
M_A = M*(1 - 3*k^2)/2'kN·m
'Shear
V = 3*M*(1 - k^2)/(2*l)'kN
'Bending at mid-span
M_a = V*a - M_A'kN·m
M_b = V*b'kN·m
'<p><b>Diagrams</b></p>
#hide
PlotWidth = 600
PlotHeight = 150
#show
'Calculate internal forces at'x_1 = ?'m
#pre

#post
'Bending
M(x) = M_A - V*x + M*(x > a)
$Plot{-M(x) @ x = 0 : l}
M(x_1)'kN·m
'Shear
V(x) = -V
$Plot{V(x) @ x = 0 : l}
V(x_1)'kN
#show
'</div>10   10  6   1
Rendered Output:
propped-cantilever-beam-concentrated-moment.png

Input data

Beam Length - l = 10  m

Load - M = 10  kN·m

Distances - a = 6  m,

b = l − a = 10 − 6 = 4 m, k = bl = 410 = 0.4

Internal forces

Bending at support

MA = M ·  ( 1 − 3 · k2 ) 2 = 10 ·  ( 1 − 3 · 0.42 ) 2 = 2.6 kN·m

Shear

V = 3 · M ·  ( 1 − k2 ) 2 · l = 3 · 10 ·  ( 1 − 0.42 ) 2 · 10 = 1.26 kN

Bending at mid-span

Ma = V · a − MA = 1.26 · 6 − 2.6 = 4.96 kN·m

Mb = V · b = 1.26 · 4 = 5.04 kN·m

Diagrams

Calculate internal forces at x1 = 1  m

Bending

M ( x )  = MA − V · x + M ·  ( x > a ) 

Plot

M  ( x1 )  = M  ( 1 )  = 1.34 kN·m

Shear

V ( x )  = -V

Plot

V  ( x1 )  = V  ( 1 )  = -1.26 kN

Linearly Distributed Load

Propped cantilever under a trapezoidal load defined by end intensities \(q_1\) and \(q_2\), locating the in-span peak from the zero of the shear \(V(x)\).

Code:
'<div style="max-width:180mm;">
#pre
'<img style="width:210pt;" class="side" alt="propped-cantilever-beam-distributed-load-linear-left.png" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load-linear-left.png">
#show
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'q_1 = ?'kN/m,'q_2 = ?'kN/m
#post
'<div class="side">
#hide
#if q_1 < q_2
    #post
    '<img style="width:210pt;" alt="propped-cantilever-beam-distributed-load-linear-right.png" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load-linear-right.png">
    #hide
#else if q_1 > q_2
    #post
    '<img style="width:210pt;" alt="propped-cantilever-beam-distributed-load-linear-left.png" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load-linear-left.png">
#else
    '<img style="width:210pt;" alt="propped-cantilever-beam-distributed-load.png" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load.png">
#end if
'</div>
Δq = q_2 - q_1'kN/m
'<p><b>Internal forces</b></p>
'Bending at support
M_A = (8*q_1 + 7*q_2)*l^2/120'kN·m
'Shear
V_A = (16*q_1 + 9*q_2)*l/40'kN
V_B = (4*q_1 + 11*q_2)*l/40'kN
'Bending at mid-span
#if '<!--'q_1 ≡ q_2'-->
    x_max = 5*l/8'm
#else
    x_max = l*(sqr(q_1^2 + 2*V_A*Δq/l) - q_1)/Δq'm
#end if
M_max = V_A*x_max - x_max^2*(3*l*q_1 + Δq*x_max)/(6*l) - M_A'kN·m
'<p><b>Diagrams</b></p>
#hide
PlotWidth = 600
PlotHeight = 150
#show
'Calculate internal forces at'x_1 = ?'m
#pre



#post
q(x) = q_1 + Δq*x/l
'Bending
M(x) = -M_A + V_A*x - (2*q_1 + q(x))*x^2/6
$Plot{-M(x) @ x = 0 : l}
M(x_1)'kN·m
'Shear
V(x) = V_A - (q_1 + q(x))*x/2
$Plot{V(x) @ x = 0 : l}
V(x_1)'kN
#show
'</div>10   5   10  1
Rendered Output:

Input data

Beam Length - l = 10  m

Load - q1 = 5  kN/m, q2 = 10  kN/m

propped-cantilever-beam-distributed-load-linear-right.png

Δq = q2 − q1 = 10 − 5 = 5 kN/m

Internal forces

Bending at support

MA =  ( 8 · q1 + 7 · q2 )  · l2120 =  ( 8 · 5 + 7 · 10 )  · 102120 = 91.67 kN·m

Shear

VA =  ( 16 · q1 + 9 · q2 )  · l40 =  ( 16 · 5 + 9 · 10 )  · 1040 = 42.5 kN

VB =  ( 4 · q1 + 11 · q2 )  · l40 =  ( 4 · 5 + 11 · 10 )  · 1040 = 32.5 kN

Bending at mid-span

xmax = l · (  q12 + 2 · VA · Δql − q1)Δq = 10 · (  52 + 2 · 42.5 · 510 − 5)5 = 6.43 m

Mmax = VA · xmax − xmax2 ·  ( 3 · l · q1 + Δq · xmax ) 6 · l − MA = 42.5 · 6.43 − 6.432 ·  ( 3 · 10 · 5 + 5 · 6.43 ) 6 · 10 − 91.67 = 56.09 kN·m

Diagrams

Calculate internal forces at x1 = 1  m

q ( x )  = q1 + Δq · xl

Bending

M ( x )  = -MA + VA · x −  ( 2 · q1 + q  ( x )  )  · x26

Plot

M  ( x1 )  = M  ( 1 )  = -51.75 kN·m

Shear

V ( x )  = VA −  ( q1 + q  ( x )  )  · x2

Plot

V  ( x1 )  = V  ( 1 )  = 37.25 kN

Partially Distributed Load

Propped cantilever with a uniform load \(q\) acting on a stretch of length \(b\) between offsets \(a\) and \(c = l - a - b\) from the supports.

Code:
'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="propped-cantilever-beam-distributed-load-p.png" class="side" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load-p.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'q = ?'kN/m
'Distances -'a = ?'m,'b = ?'m
#post
c = l - a - b'm,
d = b + c'm, 'e = a + b'm
'<p><b>Internal forces</b></p>
'Bending at support
M_A = q*((l - c^2/l)^2 - (l - d^2/l)^2)/8'kN·m
'Shear
V_A = q*b/l*(b/2 + c) + M_A/l'kN
V_B = q*b - V_A'kN
'Bending in mid-span
x_max = V_A/q + a'm
M_max = V_A*x_max - (x_max - a)^2*q/2 - M_A'kN·m
'<p><b>Diagrams</b></p>
#hide
PlotWidth = 600
PlotHeight = 150
#show
'Calculate internal forces at'x_1 = ?'m
#pre

#post
'Bending
M(x) = -M_A + V_A*x - q*(x - a)^2/2*(x > a) + q*(x - e)^2/2*(x > e)
$Plot{-M(x) @ x = 0 : l}
M(x_1)'kN·m
'Shear
V(x) = V_A - q*(x - a)*(x > a) + q*(x - e)*(x > e)
$Plot{V(x) @ x = 0 : l}
V(x_1)'kN
#show
'</div>10   5   2   6   1
Rendered Output:
propped-cantilever-beam-distributed-load-p.png

Input data

Beam Length - l = 10  m

Load - q = 5  kN/m

Distances - a = 2  m, b = 6  m

c = l − a − b = 10 − 2 − 6 = 2 m,

d = b + c = 6 + 2 = 8 m, e = a + b = 2 + 6 = 8 m

Internal forces

Bending at support

MA = q · ((l − c2l)2 − (l − d2l)2)8 = 5 · ((10 − 2210)2 − (10 − 8210)2)8 = 49.5 kN·m

Shear

VA = q · bl · (b2 + c) + MAl = 5 · 610 · (62 + 2) + 49.510 = 19.95 kN

VB = q · b − VA = 5 · 6 − 19.95 = 10.05 kN

Bending in mid-span

xmax = VAq + a = 19.955 + 2 = 5.99 m

Mmax = VA · xmax −  ( xmax − a ) 2 · q2 − MA = 19.95 · 5.99 −  ( 5.99 − 2 ) 2 · 52 − 49.5 = 30.2 kN·m

Diagrams

Calculate internal forces at x1 = 1  m

Bending

M ( x )  = -MA + VA · x − q ·  ( x − a ) 22 ·  ( x > a )  + q ·  ( x − e ) 22 ·  ( x > e ) 

Plot

M  ( x1 )  = M  ( 1 )  = -29.55 kN·m

Shear

V ( x )  = VA − q ·  ( x − a )  ·  ( x > a )  + q ·  ( x − e )  ·  ( x > e ) 

Plot

V  ( x1 )  = V  ( 1 )  = 19.95 kN

Uniformly Distributed Load

Full-span uniform load \(q\) on a propped cantilever, giving the textbook results \(M_A = q l^2 / 8\), \(V_A = 5 q l / 8\) and \(M_{max} = 9 q l^2 / 128\) at \(x = 5 l / 8\).

Code:
'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="propped-cantilever-beam-distributed-load.png" class="side" src="../../Images/mechanics/beams/propped-cantilever-beam-distributed-load.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'q = ?'kN/m
#post
'<p><b>Internal forces</b></p>
'Bending
M_A = q*l^2/8'kN·m
x_max = 5*l/8'm
M_max = 9*q*l^2/128'kN·m
'Shear
V_A = 5*q*l/8'kN
V_B = 3*q*l/8'kN
'<p><b>Diagrams</b></p>
#hide
PlotWidth = 600
PlotHeight = 150
#show
'Calculate internal forces at'x_1 = ?'m
#pre


#post
'Bending
M(x) = -M_A + V_A*x - q*x^2/2
$Plot{-M(x) @ x = 0 : l}
M(x_1)'kN·m
'Shear
V(x) = V_A - q*x
$Plot{V(x) @ x = 0 : l}
V(x_1)'kN
#show
'</div>10   5   1
Rendered Output:
propped-cantilever-beam-distributed-load.png

Input data

Beam Length - l = 10  m

Load - q = 5  kN/m

Internal forces

Bending

MA = q · l28 = 5 · 1028 = 62.5 kN·m

xmax = 5 · l8 = 5 · 108 = 6.25 m

Mmax = 9 · q · l2128 = 9 · 5 · 102128 = 35.16 kN·m

Shear

VA = 5 · q · l8 = 5 · 5 · 108 = 31.25 kN

VB = 3 · q · l8 = 3 · 5 · 108 = 18.75 kN

Diagrams

Calculate internal forces at x1 = 1  m

Bending

M ( x )  = -MA + VA · x − q · x22

Plot

M  ( x1 )  = M  ( 1 )  = -33.75 kN·m

Shear

V ( x )  = VA − q · x

Plot

V  ( x1 )  = V  ( 1 )  = 26.25 kN

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