Built in Beams II

Deflection is a term that is used to describe the degree to which a structural element is displaced under a load.

A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.

Shear and Deflection formulae for Built in and Continuous Beams

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Fixed at Both Ends. Uniform Load. Total Load W

The stress at any Point : $S = \displaystyle\frac{W\;l}{2\;Z}\left\{\displaystyle\frac{1}{6} - \displaystyle\frac{x}{l} + \left(\displaystyle\frac{x}{l} \right)^2\right\}$

The Maximum Stress is at the ends and is $\hat{S} = \displaystyle\frac{W\;l}{12\;Z}$

The Stress is zero at $\displaystyle x = 0.7887\;l$ and $x = 0.2113\;l$

The Greatest negative Stress is at the centre and is $\displaystyle S_c\;= - \frac{W\;l}{24\;Z}$

The Deflection at any Point is given by $y = \displaystyle\frac{W\;x^2}{24\;E\:I\;l}\;\;(l - x)^2$

The Maximum Deflection is at the Centre and is $\hat{y} = \displaystyle\frac{W\;l^3}{384\;E\;I}$

Fixed at Both Ends. Load at Centre.

The Stress between each end and the Load: $S = \displaystyle\frac{W}{2\;Z}\;\;\left(\displaystyle\frac{1}{4}\;l - x \right)$

The Stress at each end is : $S_e = \displaystyle\frac{W\;l}{8\;Z}$

The Stress at the middle is: $S_m\;= - \displaystyle\frac{W\;l}{8\;Z}$

These are the maximum Stresses and are equal and opposite.

The stress is zero at $\displaystyle x = \frac{1}{4}\;l$

The Deflection at any point is given by: $y = \displaystyle\frac{W\;x^2}{48\;E\;I}\;\;(3l - 4x)$

The Maximum Deflection is at the Load and is: $\displaystyle \hat{y} = \frac{W\;l^3}{192\;E\;I}$

Fixed at Both Ends. Load at any Point.

The Stress at any Point within the segment of length $a$ :

$S_a = \frac{W\;b^2}{Z\;l^3}\;\;\left(al - x[l + 2a] \right)$

The Stress at any point within the segment of length $b$ :

$S_b = \frac{W\;a^2}{Z\;l^3}\;\;\left(bl - v[l + 2b] \right)$

The Stress at the end next to segment of length $a$ : $S = \displaystyle\frac{W\;a\;b^2}{Z\;l^2}$

The Stress at the end next to segment of length $b$ : $S = \displaystyle\frac{W\;a^2\;b}{Z\;l^2}$

The Maximum Stress is at the end next to the shorter segment.

The Stress is Zero for: $x = \displaystyle\frac{a\;l}{l + 2\;a}$ and $v = \displaystyle\frac{b\;l}{l + 2\;b}$

The Greatest negative stress is at the Load and is given by: $S\;= - \displaystyle\frac{2\;W\;a^2\;b^2}{Z\;l^3}$

The Deflection for the segment of length $a$ is given by:

$y = \frac{W\;x^2\;b^2}{6\;E\;I\;l^3}\;\;2a(l - x) + l(a - x)$

The Deflection for the segment of length $b$ is given by:

$y = \frac{W\;v^2\;a^2}{6\;E\;I\;l^3}\;\;2b(l - v) + l(b - v)$

The deflection at the Load is: $y = \displaystyle\frac{W\;a^3\;b^3}{3\;E\;I\;l^3}$

Let $b$ be the length of the longer segment and $a$ the shorter one.

The Maximum Deflection is in the longer Segment and occurs at $\displaystyle v = \frac{2\;b\;l}{l + 2\;b}$

$\hat{y} = \frac{2\;W\;a^2\;b\63}{3\;E\;I\;(l + 2\;b)}$

Continuous Beam with two Equal Spans. Uniform Load

The Stress at any Point is given by: $s = \displaystyle\frac{W\;(l - x)}{2\;Z\;l}\;\;\left(\displaystyle\frac{1}{4}l - x \right)$

The Maximum Stress is at Point $A$ and is: $\hat{S} = \displaystyle\frac{W\;l}{8\;Z}$

The Stress is zero at $\displaystyle x = \frac{1}{4}l$

The greatest negative Stress is at $\displaystyle x = \frac{5}{8}\;l$ and is: $y\;= - \displaystyle\frac{9}{128}\;\;\displaystyle\frac{W\;l}{Z}$

The Deflection at any Point is given by: $y = \displaystyle\frac{W\;x^2\;(l - x)}{48\;E\;I\;l}\;\;(3l - 2x)$

The Maximum Deflection is at $\displaystyle x = 0.5785\;l$ and is given by: $\hat{y} = \displaystyle\frac{W\;l^3}{185\;E\;I}$

The Deflection at the centre of each Span is: ${y} = \displaystyle\frac{W\;l^3}{192\;E\;I}$

The Deflection at the point of greatest negative Stress is ${y} = \displaystyle\frac{W\;l^3}{187\;E\;I}$
And is at $x = \displaystyle\frac{5}{8}\;l$

A Continuous Beam with Two unequal Spans and unequal uniform Loads.

Between $\displaystyle R_1$ and $R$ The Stress is given by: $S = \displaystyle\frac{l_1 - x}{Z}\left\{\displaystyle\frac{(l_1 - x)\;W_1 }{2\;l_1} - R_1\right\}$

Between $\;R_2$ and $R$ The Stress is given by: $S = \displaystyle\frac{l_2 - u}{Z}\;\left\{\displaystyle\frac{(l_2 - u)\;W_2}{2\;l_2} - R_2\right\}$

Stress at the Support $R$ is: $S_R = \displaystyle\frac{W_1\;l_1^2 + W_2\;l_2^2}{8\;Z\;(l_1 + l_2)}$

The greatest Stress in the first span is at: $x = \displaystyle\frac{l_1}{W_1}\;\;(W_1 - R_1)$

And is $\displaystyle \frac{R_1^2\;l_1}{2\;Z\;W_1}$

The greatest Stress in the second Span is at: $u = \displaystyle\frac{l_2}{W_2}\;\;(W_2 - R_2)$

And is $\displaystyle - \frac{R_2^2\;l_2}{2\;Z\;W_2}$

The Deflection between $R_1 \;and\; R$ is given by:

$y = \frac{x(l_1 - x)}{24\;E\;I}\left\{(2l_1 - x)(4R_1 - W_1) - \frac{W_1\;(l_1 - x)^2}{l_1} \right\}$

The Deflection between $R_2\; and \;R$ is given by:

$y = \frac{u(l_2 - u)}{24\;E\;I}\left\{(2l_2 - u)(4R_2 - W_2) - \frac{W_2\;(l_2 - u)^2}{l_2} \right\}$

The above example is so complicated that convenient general expressions for the maximum deflections cannot be obtained

Between point $A$ and the Load the Stress at any Point is: $S = \displaystyle\frac{W}{16\;Z}\;\;(3\;l - 11\;x)$

Between Point $B$ and the Load the stress at any Point $S\;= - \displaystyle\frac{5}{16}\;\;\displaystyle\frac{W\;v}{Z}$

The Maximum Stress at Point $A$ $\hat{S} = \displaystyle\frac{3}{16}\;\;\displaystyle\frac{W\;l}{Z}$

The Stress is Zero at: $x = \displaystyle\frac{3}{11}\;l$

The greatest negative Stress is at the centre of the Span and is:

$\hat{S}_{-ve}\;= - \frac{5}{32}\;\;\frac{W\;l}{Z}$

Between Point $A$ and the Load the Deflection at any point is:

$y = \frac{W\;x^2}{96\;E\;I}\;\;(9\;l - 11\;x)$

Between Point $B$ and the Load the Deflection at any point is:

$y = \frac{W\;v}{96\;E\;I}\;\;(3\;l^2 - 5\;v^2)$

The Maximum deflection is at $\displaystyle v = 0.4472$ and is: $\hat{y} = \displaystyle\frac{W\;l^3}{107.33\;E\;I}$

The Deflection at the Load is: $y_w = \displaystyle\frac{7}{768}\;\;\displaystyle\frac{W\;l^3}{E\;I}$

A Continuous Beam with Two unequal Spans with two unequal Loads at any point on each

Between $\displaystyle R_1$ and $W_1$ the Stress is: $S\;= - \displaystyle\frac{w\;r_1}{Z}$

Between $\displaystyle R$ and $W_1$ the Stress is: $S = \displaystyle\frac{I}{l_1\;Z}\;\;[m(l_1 - u) - W_1\;a_1\;u]$

Between $\displaystyle R$ and $W_2$ the Stress is: $S = \displaystyle\frac{I}{l_2\;Z}\;\;[m(l_2 - x) - W_2\;a_2\;x]$

Between $\displaystyle R_2$ and $W_2$ the Stress is: $S\;= - \displaystyle\frac{v\;r_2}{Z}$

Stress at the Load: $S\;= - \displaystyle\frac{a_1\;r_1}{Z}$

Where $m = \frac{1}{2(l_1 + l_2)}\;\;\left(\frac{w_1\;a_1\;b_1}{l_1}\;(l_1 + a_1) + \frac{w_2\;a_2\;b_2}{l_2}\;\;(l_2 + a_2) \right)$

Stress at Support $R$ : $S_R = \displaystyle\frac{m}{Z}$

Stress at load $W_2$ : $S\;= - \displaystyle\frac{a_2\;r_2}{Z}$

The greatest of these is the maximum Stress.

The Deflection between $\displaystyle R_1$ and $W_1$

$y = \frac{w}{6\;E\;I}\;\left\{(l_1 - w)(l_1 + w)r_1 - \frac{W_1\;b_1^3}{l_1} \right\}$

The Deflection between $\displaystyle R$ and $W_1$

$y = \frac{u}{6\;E\;I\;l_1}\;\;[W_1\;a_1\;b_1(l_1 + a_1) - W_1\;a_1\;u^2 - m(2\;l_1 - u)(l_1 - u)]$

The Deflection between $\displaystyle R$ and $W_2$

$y = \frac{x}{6\;E\;I\;l_1}\;\;[W_2\;a_2\;b_2(l_2 + a_2) - W_2\;a_2\;x^2 - m(2\;l_2 - x)(l_2 - x)]$

The Deflection between $\displaystyle R_2$ and $W_2$

$y = \frac{v}{6\;E\;I}\;\left\{(l_2 - v)(l_2 + v)r_2 - \frac{W_2\;b_2^3}{l_2} \right\}$

The Deflection at Load $\displaystyle W_1$ : $y = \displaystyle\frac{a_1\;b_1}{6\;E\;I\;}\;[2a_1\;b_1\;W_1 - m(l_1 + a_1)]$

The Deflection at Load $\displaystyle W_2$ : $y = \displaystyle\frac{a_2\;b_2}{6\;E\;I\;}\;[2a_2\;b_2\;W_2 - m(l_2 + a_2)]$