An engineer is currently tasked with designing a steel beam which is to support a flooring system above it$.$

The weight of the flooring system will act as a distributed load over the length of the beam, which in turn will

cause the beam to bend, and deflect. It is your task to check that the cross$-$section for this beam below is

suitable subject to the design constraints given: both "Strength" and "Serviceability".

Satisfying strength ensures that stresses within the structural or mechanical member do not exceed critical values that

would cause them to fail, or exhibit other undesirable mechanical responses. In this case, we wish to ensure that no part of

the material reaches yield. Satisfying serviceability constraints ensure that deformations of a structural or mechanical

member do not exceed critical values that would prevent them from performing their desired function. In this case, we wish

to minimise any deflections of the beam so that the floor above it does not 'sag'

The member can be modelled as a simply supported beam carrying a UDL of $w=11k\frac{N}{m}$ across its

span.

The length of the beam between supports is $L=4m.$ The Young's modulus of the material is $E=200$

GPa

The beam rests on supporting joists, which may be modelled as roller supports in our analysis as they only

constrain motion of points A and $B$ in the vertical direction. Since there are no lateral loads acting on the

system, equilibrium of horizontal forces is already satisfied.

Limits of Design

Firstly, we calculate the limits for our design which need to be satisfied.

Strength

The yield stress of the steel material is ${}_Y=250$MPa. In this case, we will design strength according to a factor of safety

of $FS=2.$

Therefore, the maximum allowable stress in the material is given by:

${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=\frac{{}_Y}{FS}=,$MPa

The factor of safety ensures that the design of the material is such that it does not come close to reaching its critical value.Serviceability.

The following is an extract from Table C$.1$ in the Australian$/$New Zealand Standards for Structural Design

actions $($AS$1170\mathrm{.}0)$:

This extract from the table provides limits for deflections for various structural components to satisfy

serviceability. In this case, we are interested in the row related to "Floor joists$/$Beams$",$ in order to reduce the

occurrence of 'sagging' of the beam.

Therefore we need to ensure the deflection of the mid$-$span of the beam is below "Span$/300",$ where span

refers to the length of the beam between supports.

$v_{\text{a}\text{l}\text{l}\text{o}\text{w}}=\frac{L}{300}=,mm$

Loading of Member

Using equilibrium, calculate the two vertical support reactions acting on the beam:

$A_y=,kN(assume$ upwards $is$ positive$)$

$By=,kN(assume$ upwards $is$ positive$)$

Give an expression for the internal bending moment of the beam as a function of $x$ along the beam. Please

take $x$ as positive from left to right, with origin at $A.$

$M(x)=$

$kNm($use the positive sign convention for bending

moment taught in class$)$

The only unknown in your expression should be $x.$ Please enter exact known values of all other parameters

$($use fractions rather than decimals$)$Strength Analysis

The cross$-$section of the beam here is the $"$I$-$section" shown below. The dimensions are given as:

$b=160mm$

$d=220mm$

$t=15mm$

With reference to the origin shown above, what is the vertical coordinate of the centroid for this cross$-$section:

$\frac{?}{\text{b}\text{a}\text{r}}(y)=,mm$

What is the second moment of area about the horizontal axis passing through the centroid:

$I_{x^{\text{'}}{x}^{\text{'}}}=,m{m}^4$

For a linear elastic material, the depth of the centroid is also the depth of the neutral axis.

Calculate the maximum internal bending moment within the beam $($this occurs at the midspan$)$:

$M_{\text{m}\text{a}\text{x}}=$

What is the largest magnitude of stress due to bending acting on the cross$-$section $($calculate stress at

extreme points of cross$-$section $-$ top and bottom fibres $-$ and then give magnitude of largest stress$)$:

${}_{\text{m}\text{a}\text{x}}=$

MPa

Does this stress satisfy the design constrain for strength? ${}_{\text{m}\text{a}\text{x}}<msub><mrow><mi></mi></mrow><mrow><mtext>a</mtext><mtext>l</mtext><mtext>l</mtext><mtext>o</mtext><mtext>w</mtext><mtext></mtext></mrow></msub>$Serviceability Analysis

Next, calculate the deflection of the beam at the mid$-$span due to the loads using the "integration" approach

taught in class. The cross$-$section is the l$-$beam shown earlier.

Earlier you determin