Fully Restrained Beams¶

CalcpadCE worksheets in this section treat the fully restrained beam — a single span clamped against rotation and translation at both ends. The two end fixities make the system twice statically indeterminate, so each load case produces a pair of fixed-end moments at the supports along with the shears and the in-span bending peak.

The point force and point moment cases give the textbook fixed-end formulas \(M_A = F a b^2 / l^2\) and \(M_B = F a^2 b / l^2\) (and their moment-load counterparts), useful as inputs to slope-deflection and moment-distribution analyses. The distributed cases cover the uniform load with \(M_A = M_B = q l^2 / 12\), the linearly varying load split between the two supports as \((3 q_1 + 2 q_2) l^2 / 60\) and \((2 q_1 + 3 q_2) l^2 / 60\), and a partial uniform load of length \(b\) between offsets \(a\) and \(c\).

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¶

Fixed-end moments \(M_A = F a b^2 / l^2\) and \(M_B = F a^2 b / l^2\) for a clamped-clamped beam carrying a point force \(F\) at distance \(a\) from the left support.

Code:

'<div style="max-width:180mm">
'<img style="width:210pt;" alt="fixed-beam-concentrated-force.png" class="side" src="../../Images/mechanics/beams/fixed-beam-concentrated-force.png">
'<p><b>Input data</b></p>
'Beam Length -'l=?'m
'Load -'F=?'kN
'Distances -'a=?'m,'b=l-a'm
#post
'<p><b>Internal forces</b></p>
'Bending at supports
M_A = F*a*b^2/l^2'kN·m
M_B = F*a^2*b/l^2'kN·m
'Bending at mid-span
M_max = 2*F*a^2*b^2/l^3'kN·m
'Shear
V_A = F*b^2/l^2*(1+2*a/l)'kN
V_B = F*a^2/l^2*(1+2*b/l)'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  2   1

Concentrated Moment¶

Fully restrained beam loaded by a concentrated moment \(M\) at distance \(a\), with a constant shear \(V = 6 M a b / l^3\) and a step in the bending diagram across the load.

Code:

'<div style="max-width:180mm">
'<img class="side" style="width:210pt;" alt="fixed-beam-concentrated-moment.png" src="../../Images/mechanics/beams/fixed-beam-concentrated-moment.png">
'<p><b>Input data</b></p>
'Beam length -'l = ?'m
'Load -'M = ?'kN·m
'Distances -'a = ?'m,'b = l - a'm
#post
'<p><b>Internal forces</b></p>
'Bending at supports
M_A = M*b*(2 - 3*b/l)/l'kN·m
M_B = M*a*(2 - 3*a/l)/l'kN·m
'Shear
V = 6*M*a*b/l^3'kN
'Bending at mid-span
M_a = V*a - M_A'kN·m
M_b = V*b - M_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  4   1

Linearly Distributed Load¶

Trapezoidal load between intensities \(q_1\) and \(q_2\), with the support moments split as \((3 q_1 + 2 q_2) l^2 / 60\) and \((2 q_1 + 3 q_2) l^2 / 60\).

Code:

'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="fixed-beam-distributed-load-linear.png" class="side" src="../../Images/mechanics/beams/fixed-beam-distributed-load-linear.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'q_1 = ?'kN/m,'q_2 = ?'kN/m
Δq = q_2 - q_1'kN/m
#post
'<p><b>Internal forces</b></p>
'Bending at supports
M_A = (3*q_1 + 2*q_2)*l^2/60'kN·m
M_B = (2*q_1 + 3*q_2)*l^2/60'kN·m
'Shear
V_A = l/20*(7*q_1 + 3*q_2)'kN
V_B = l/20*(3*q_1 + 7*q_2)'kN
'Bending at mid-span
#if '<!--'Δq ≡ 0'-->'
    x_max = l/2'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 - (3*l*q_1 + Δq*x_max)*x_max^2/(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

Partially Distributed Load¶

Uniform load \(q\) over a stretch of length \(b\) between offsets \(a\) and \(c = l - a - b\), with a guard against an invalid configuration when \(a + b > l\).

Code:

'<div style="max-width:180mm;">
'<img style="width:210pt;" alt="fixed-beam-distributed-load-partial.png" class="side" src="../../Images/mechanics/beams/fixed-beam-distributed-load-partial.png">
'<p><b>Input data</b></p>
'Beam Length -'l = ?'m
'Load -'q = ?'kN/m
'Distances -'a = ?'m,'b = ?'m,'c = l - a - b'm
'<!--
#if c < 0
    '<-->
    'It must be satisfied that <i>a</i> + <i>b</i> &le; <i>l</i>
    '<!--
#else
    '<-->
    #post
    d = a + b'm,'e = b + c'm
    '<p><b>Internal forces</b></p>
    'Bending at supports
    M_A = q*(e^3*(l + 3*a) - c^3*(l + 3*d))/(12*l^2)'kN·m
    M_B = q*(d^3*(l + 3*c) - a^3*(l + 3*e))/(12*l^2)'kN·m
    'Shear
    V_A = (q*b*(b/2 + c) + (M_A - M_B))/l'kN
    V_B = (q*b*(b/2 + a) + (M_B - M_A))/l'kN
    'Bending at mid-span
    x_max = V_A/q + a'm
    M_max = V_A*(x_max + a)/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 - d)^2/2*(x > d)
    $Plot{-M(x) @ x = 0 : l}
    M(x_1)'kN·m
    'Shear
    V(x) = V_A - q*(x - a)*(x > a) + q*(x - d)*(x > d)
    $Plot{V(x) @ x = 0 : l}
    V(x_1)'kN
    #show
    '<!--
#end if
'<-->
'</div>10   5   1   6   1

Uniformly Distributed Load¶

Full-span uniform load \(q\), recovering \(M_A = M_B = q l^2 / 12\), \(V_A = V_B = q l / 2\) and the mid-span \(M_{max} = q l^2 / 24\).

Code:

'<div style="max-width:180mm">
'<img style="width:210pt;" alt="fixed-beam-distributed-load-uniform.png" class="side" src="../../Images/mechanics/beams/fixed-beam-distributed-load-uniform.png">
'<p><b>Input data</b></p>
'Beam Length -'l=?'m
'Load -'q=?'kN/m
#post
'<p><b>Internal forces</b></p>
'Bending at supports
'<i>M</i><sub>A</sub>='M_B = q*l^2/12'kN·m
'Bending at mid-span
M_max = q*l^2/24'kN·m
'Shear
'<i>V</i><sub>A</sub>='V_B=q*l/2'kN
'<p><b>Diagrams</b></p>
#hide
M_A = M_B
V_A = V_B
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

Spotted an error? Edit these examples.