In the Beams and frames pathway, we read bending moment diagrams and checked them against the standard formulas. Now we'll take the next step and turn those internal actions into stresses. We'll start with the bending stress in a simply supported beam. We already know what the answer should be, so it's a good place to learn how Structure Bench calculates and displays stress.
In this step, we'll paint the normal stress over a simply supported UB, read it at midspan and check it by hand.
Normal stress in a 6 m beam. Orbit, read the results and hover the members; nothing you do here changes the model.
Members can copy this model into their own projects and work the step in it. Join or sign in.
1.0 From bending moment to normal stress
A normal stress acts perpendicular to the face of a cross-section - in other words, along the length of the member.
An analysis tells us the bending moment at every point along a beam. To turn that moment into a stress, beam theory makes one key assumption: plane sections remain plane. This simply means that a cross-section that's flat before the beam bends stays flat afterwards - it just rotates. So, the strain varies linearly through the depth of the section. It's zero at the neutral axis (the line through the section where the fibres neither lengthen nor shorten) and largest at the top and bottom. While the steel stays elastic, stress is proportional to strain, so the stress varies linearly too,
For a symmetric section like a UB, the neutral axis sits at mid-depth and the largest stress occurs at the extreme fibres, a distance h/2 away. We usually fold I and h/2 together into a single section property, the elastic section modulus,
Wel​=h/2I​,σmax​=Wel​M​
In a sagging beam, the top fibres shorten and the bottom fibres lengthen, so we get compression on top and tension underneath. In a hogging region, it's the other way round. Don't get these mixed up when you read a stress plot - the sign of the bending moment tells you which face is in tension.
2.0 The model
Our model is a UB 406 x 178 x 60 in S355 steel. It spans 6 m between a pin and a roller and is subject to a uniformly distributed load of 30 kN/m. The midspan moment is the one we all know,
M=8wL2​=830×62​=135 kN m
The beam is statically determinate, so this comes from statics alone. The section only enters when we convert the moment into a stress.
The model is already solved, so its results are waiting in the panel on the left, under Display layers.
Things to try
Press Load the model and switch on the Bending moment layer. Hover over midspan (with Select active in the main menu) and read the moment. This is the 135 kN m we calculated above.
Switch the bending moment off, switch on Member stress and select the σ chip. The beam is now painted with the normal stress through its depth, and the colour bar shows the scale.
Hover over the beam at midspan, then near a support. The card shows the forces at that point and the largest stress through the depth. Notice how the stress falls as you move towards the support, in step with the bending moment.
Switch to the U chip and hover at midspan again to read the utilisation. U is the von Mises (equivalent) stress divided by the steel's yield strength. At midspan the shear force is zero, so the von Mises stress is just the normal stress we've been looking at.
3.0 What you'll see
What you should see
Where
Moment
σ, largest through the depth
Midspan
135.0 kN·m
127.0 MPa (tension, at the bottom fibre)
At a support
0
0
The card gives one stress at the point you hover: the largest through the depth at that point, marked peak. At midspan, the two ends of the colour bar are the top and bottom fibres, −127.0 and +127.0 MPa.
With the U chip, the midspan fibres read a utilisation of about 0.36.
The painted beam shows the behaviour we described in section 1.0. Through the depth, the colour runs from cold at the top (compression) to warm at the bottom (tension), passing through zero at mid-depth, which is the neutral axis. Along the span, the colours fade towards the supports, following the parabola of the bending moment diagram down to zero.
Structure Bench is doing exactly what we'd do by hand. At every point along the member, it takes the internal forces from the analysis and evaluates the beam theory formula over the cross-section. Positive is tension and negative is compression, so the top fibre is at −127 MPa and the bottom fibre at +127 MPa. They're equal in size because the section is symmetric.
Now, the U chip. Strictly, it's the von Mises (equivalent) stress, σvM​=σ2+3τ2​, divided by the yield strength, fy​ = 355 MPa for the S355 in this model, with the partial factor γM0​ taken as 1.0. At midspan there's no shear force, so σvM​ is simply σ and the beam is at about 36% of yield (127 / 355 = 0.36). Hover near a support and you'll see U doesn't drop to zero with the moment - the shear stress is doing the work there instead. Notice that the colour scale is fixed: it always runs from 0 to 1, so the same colour means the same thing in every model.
Keep in mind that this is an elastic cross-section check and nothing more. A real beam design asks more than that - lateral-torsional buckling, for example, can govern a long, unrestrained beam well before its extreme fibre yields. For more on this, take a look at this EngineeringSkills tutorial on steel beam design.
Hand check for the midspan stress
The section modulus. From the section tables, the UB 406 x 178 x 60 has I = 21,600 cm4 and h = 406.4 mm. The extreme fibre sits half the depth away from the neutral axis, so, working in centimetres,
Wel​=40.64/221,600​=1,063 cm3
(the tables list 1,060 cm3, rounded).
The midspan stress. Working in N mm and mm3, the stress comes out directly in MPa,
σ=Wel​M​
This agrees with the painted 127.0 MPa. The moment is sagging, so the stress is compression on top and tension underneath, matching the signs on the colour bar.
4.0 What the stress view assumes
What the view assumes
The member stress view is beam theory, evaluated on the idealised section. It doesn't see holes, connections, stiffeners or the concentrated stresses around a support or a point load. In other words, it's the stress you'd calculate by hand, evaluated at every point along the member.
In a real beam, those local details are often where the trouble is: the web above a bearing, or a bolt hole in a tension flange. So, use the painted stresses for the member as a whole, and check bearings, holes and connections separately.
5.0 What to take from this step
We've seen that the normal stress from bending varies linearly through the depth of a section. It's zero at the neutral axis and largest at the extreme fibres, and along the member it follows the bending moment diagram. The member stress view paints exactly that, so you can read the stress anywhere and check it with M/Wel​.
Whenever you look at a stress plot, it's worth doing what we did here: read the moment at the most highly stressed point and divide by the section modulus. If the numbers don't agree, something in the model (most often the section or the load) isn't what you think it is.
Next, we'll add an axial force and bend a column about both of its axes, and watch what happens to the neutral axis.