SE $110$ A $-$ Homework $8$ Due: $8$:$15$pm May $31,2024$

Topic: Beam deflection, shear and bending stress

Consider the beam shown in Fig. $1$ ; the croes section of this beam is similar to a standard $W250$

$x89$ Wide Flange Steel Beam with dimension detailed in the figure. We need to make a holistic

Figure $1$: An I$-$beam loaded by a uniformly distributed losd along one$-$half its length, $2L_{2,p}$ viz in the interval,

$E{I}_{33}\frac{d^2(x_1)}{d{x}_1^2}=M(x_1)x_1\frac{d}{d{x}_1}\{E{I}_{33}\frac{d^{2}v(x_1)}{d{x}_1^2}\}=\frac{dM(x_1)}{d{x}_1}=-V(x_1),\frac{d^2}{d{x}_1^2}\{E{I}_{33}\frac{d^{2}v(x_1)}{d{x}_1^2}\}=\frac{d^{2}M(x_1)}{d{x}_1^2}=-\frac{dV(x_1)}{d{x}_1}=q(x_1).\frac{dV}{d{x}_1}=-q(x_1)$ and

$\frac{dM}{d{x}_1}=-V(x_1)E{I}_{3s}0.$ The beam $is$ fixed within rigid walls $at$ both ende. All data $is$ shown $in$ the figure.

analysis $of$ this beam and assess deflections, bending moments, shear forces, and both bending

and shear stresser. $Itis$ best $to$ recall some background for perspective used for the overall

strategy $of$ solution.

Elements $of$ Beam theory: Recall the deflection$-$moment equation derived $in$ class, viz.

$E{I}_{33}\frac{d^2(x_1)}{d{x}_1^2}=M(x_1)$

$We$ now differentiate this with respect $to{x}_1$ again $to$ obtain

$\frac{d}{d{x}_1}\{E{I}_{33}\frac{d^{2}v(x_1)}{d{x}_1^2}\}=\frac{dM(x_1)}{d{x}_1}=-V(x_1),$

and yet again $to$ obtain

$\frac{d^2}{d{x}_1^2}\{E{I}_{33}\frac{d^{2}v(x_1)}{d{x}_1^2}\}=\frac{d^{2}M(x_1)}{d{x}_1^2}=-\frac{dV(x_1)}{d{x}_1}=q(x_1).$

Note these equations embody our earlier derived equilibrium equations

$\frac{dV}{d{x}_1}=-q(x_1)$ and

$\frac{dM}{d{x}_1}=-V(x_1)$

Now let's assuume for now that $E{I}_{3s}is$ constant along the beam and note that $in$ terms $of$ the deflection $v(x_1)$ we then have

$M(x_1)=E{I}_{33}\frac{d^{2}v(x_1)}{d{x}_1^2},$

$V(x_1)=-E{I}_{33}\frac{d^{3}v(x_1)}{d{x}_1^3},$

$E{I}_{33}\frac{d^{4}v(x_1)}{d{x}_1^1}=q(x_1).$

The reason the last of these equations is written in the orientation it has is due to our stratepy

of solution that is adopted for this analysis. Note that eqs. $5$a$,$b are equations that give shesar

force and bending moment, respectively.

Let us define the tasks and procedure.

$1$a$.$ Since the beam has no interior supports, write the expression for $q(x_1)$ for the applied

loading using singularity functions such as $(x_1-a{)}^2$ and use it to construct eq$.$ Sc$.$

$1$b$.$ Integrate the $4^{th}$ order ordinary differential equation obtained from eq$.5$c$,$ and express

the varions integration constants along the way. At each integration besr in mind eqs. $5$a$,$b

and ask yourself if you can identify values for any such constants due to knowledge of either

moments or shear forces. Note, however, you have not yet performed any pre$-$analysis via the

usaal methods of statics. Note that as you began you knew all information about the "applied

forces" and it is only the integration constants that require evaluation.

Ib$-$a$.$ Bearing in mind eqs. $5$a$,$b$,$ write down equations for shear force and bending moment as

soon as you can even though they may contain integration constants yet unknown.

$1$c$.$ Invole all boundary conditions $($be$\text{'}$s$)$ and evaluate all integration constants and construct

the shear foree and bending moment diagrams and sbetch them neatly. Identify ley points such

as maxima and minima along these diagrams. Plot those diagrams in terms of $w_0$ and $L$; you

will insert actual numbers later when evaluating streses. Note you should have boen esaluating

integration constants as you performed intogrations as scon as you werr able to$.$

Id$.$ Evaluate the complete deflection curve, i$.$e$.v(x_1),$ and sketch it in terms of the parameters,

$u_0,L,$ and $E{I}_{sst}.$ However, now put in mumbers and compute and indicate the actual maximum

deflection at the point it occurs.

$1$e$.$ Evaluate the moment of inertia for this cross section and evaluate the maximum bending

stresses, both the extreme tensile and compressive bending stresses and indicate their locations

along the beam.

If$.$ Evaluate the maximum shear stress, and indicate its location. Also, evaluate the shear

stress in the web just below the points at the top and bottom of the beam where the web

meets the flanges; these points are labeled A and $B$ in Fig. $1.$ Evaluate the shear stress at the

neutral axis of the beam. Perform all evaluations such as $Q()$ as original calculations without

appealing to canned formulas. Finally, evaluate the maximum shear stress in the flange and

indicate its location.

$1$g$.$ Compute the ratio of maximum $($i$.$e in magnitude$)$ shear stress to maximum bending

stress.