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Structural Mechanics Simulator

T-Beam Section Simulator — Properties & Bending Stress

Compute the centroid, moment of inertia, section modulus, plastic section modulus and shape factor of a T-section in real time from flange and web dimensions, with the neutral axis and bending stress visualized.

Parameters
Flange width b_f
mm
Flange thickness t_f
mm
Web thickness t_w
mm
Web height h_w
mm

Total height h = h_w + t_f. The centroid y̅ is measured downward from the top of the flange.

Results
—
Moment of inertia I
—
Neutral axis position y̅ (from top)
—
Bottom section modulus S_bot = I/(h-y̅)
—
Shape factor f = Z_p / S_min
Cross-Section and Bending Stress Distribution

Left = T cross-section (red line = elastic neutral axis y̅, yellow dashed = plastic neutral axis PNA) / Right = bending stress σ = M·y / I (top = compression, bottom = tension)

Theory & Key Formulas

A T-section is an asymmetric cross-section made of a flange b_f x t_f and a web t_w x h_w. The centroid and section properties follow from elementary mechanics of materials.

Total area and centroid y̅ (distance below the top of the flange):

$$A = b_f t_f + t_w h_w,\qquad \bar{y} = \frac{b_f t_f \cdot \tfrac{t_f}{2} + t_w h_w \cdot (t_f + \tfrac{h_w}{2})}{A}$$

Moment of inertia I about the elastic neutral axis (parallel-axis theorem):

$$I = \frac{b_f t_f^3}{12} + b_f t_f\!\left(\bar{y}-\tfrac{t_f}{2}\right)^2 + \frac{t_w h_w^3}{12} + t_w h_w\!\left(t_f + \tfrac{h_w}{2} - \bar{y}\right)^2$$

Section moduli (top and bottom) and the shape factor f:

$$S_\text{top} = \frac{I}{\bar{y}},\quad S_\text{bot} = \frac{I}{h - \bar{y}},\quad f = \frac{Z_p}{S_\text{min}}$$

The plastic neutral axis PNA is the position that splits A/2 above and below; the plastic section modulus is Z_p = (A/2)(d_t + d_b), where d_t and d_b are the distances from the PNA to the centroids of the upper and lower halves.

What is the T-Beam Section Simulator

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Beam cross-sections come not only as rectangles and I-shapes but also T-shapes? Why choose such an asymmetric form?
🎓
Good question. The T-section appears very often in reinforced-concrete slab-and-beam systems and in ribbed steel members. Putting the wide flange on the compression side fits perfectly with concrete carrying compression and the rebar carrying tension. In the simulator above, try increasing the "flange width b_f" — you will see the centroid y̅ rise sharply toward the flange.
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At the defaults the neutral axis sits 78.6 mm below the top, not at the middle of the 320 mm total height. That is quite far off-center.
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Right — the centroid is the area-weighted mean position, so it is pulled toward where the area is concentrated. The formula is $\bar{y} = (\sum A_i y_i)/\sum A_i$, the sum of each part's area times its centroid distance divided by the total area. With a flange of 200x20 = 4000 mm² and a web of 10x300 = 3000 mm², the centroid lies on the flange side.
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Oh, the section moduli S_top and S_bot are also very different. The bottom one is much smaller.
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That is the most important point about T-sections. S = I / y, where y is the distance from the neutral axis. The distance to the bottom (241 mm) is three times the distance to the top (79 mm), so S_bot is less than a third of S_top. Since the bending stress is sigma = M/S, the larger stress appears on the bottom side — the tension side. Design usually ends up governed by the tensile strength on the bottom.
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There is also a shape factor of 1.77 — what does that mean?
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The shape factor f = Z_p/S shows how much reserve remains between "the moment when the extreme fiber first yields" and "the moment when the entire section is fully plastic". A rectangle gives 1.5, an I-section 1.1 to 1.2, and a T-section 1.6 to 1.9. Asymmetry lets the plastic neutral axis shift away from the elastic one, leaving more reserve. In plastic-hinge design this reserve is precisely what matters.

Frequently Asked Questions

The centroid (elastic neutral axis) is "the position where the first moment of area is zero", whereas the plastic neutral axis (PNA) is "the position that splits the area into two equal halves". These are different definitions. They coincide for symmetric sections like rectangles, but not for asymmetric ones like T-sections. With the defaults, the centroid lies 78.6 mm and the PNA only 17.5 mm from the top — over 60 mm apart. Use the centroid for elastic behavior and the PNA for plastic behavior.
The section properties (I, Z_p, A) themselves are unchanged by flipping. However, combined with the sign of the bending moment, the utilization of the material differs between the compression and tension sides. Reinforced concrete normally has the flange on top (compression) so the slab works integrally; for steel, where strength is symmetric in tension and compression, the choice is made by stress level and buckling considerations.
A thin web contributes relatively little to I, so the moment of inertia drops only mildly. However, the web carries the shear force, so making it too thin raises the shear stress and lets web buckling or shear yielding govern. In practice you must check shear strength, web slenderness ratio for local buckling, and the need for stiffeners, in addition to flexural strength.
The pure geometric section properties are accurate, but for actual design you must add material factors (E, yield stress σ_y), effective flange width for concrete beams, safety factors, checks against shear, local and lateral-torsional buckling, fatigue and so on. The tool is well suited to learning, concept design and preliminary studies; the final design decisions must follow each country's structural codes (steel design standards, road-bridge specifications, concrete standards, etc.).

Real-World Applications

Reinforced-concrete slab-and-beam systems: The T-section is one of the most common cross-sections in reinforced-concrete structures. When the floor slab and the beam are cast monolithically, the slab acts as the compression flange of a "T-beam". The effective flange width is a key design parameter and is specified in each country's structural codes. Taking the slab contribution into account leads to far more rational reinforcement than designing the beam as a bare rectangle.

Steel beams and structural tees: Rolled T-shapes and cut tees made by slitting an H-section in half are used as truss chords, braces and lightweight beams. They suit members carrying one-directional bending or members needing connections on one side only. With the flange on one side, bolted and welded connections are easy to make.

Composite beams in buildings and bridges: A steel girder topped by a concrete slab and connected through stud shear connectors behaves, for section property purposes, as a T-section. Steel carries tension and concrete carries compression, using each material to maximum advantage. This system is widely used in bridge decks and floor systems of tall buildings.

Machine elements and stiffened structures: Ribbed machine frames, stiffeners on thin-plate structures, and rail sections are effectively T-sections with stiffening on one side. Using a flange to inflate I is a universal trick, common to machine design and building design alike.

Common Misconceptions and Cautions

The most common misconception is to assume "the top and bottom can be evaluated with the same stress check". Because a T-section is asymmetric, the top and bottom section moduli S_top and S_bot differ greatly. At the defaults, S_top is about 847 cm³ while S_bot is only 275 cm³ — over three times smaller. Under the same bending moment the bottom stress is three times the top stress, so you must always check both sides and use the smaller S as the governing criterion.

The next mistake is to confuse the centroid y̅ with the plastic neutral axis PNA. They are mathematically different quantities. They coincide for symmetric sections but not for T-sections. With the defaults, the elastic neutral axis is 78.6 mm and the PNA is 17.5 mm from the top — more than 60 mm apart. Use y̅ for elastic stress, and the distance from the PNA for the plastic moment and the plastic section modulus. In the simulator, the red line marks the elastic neutral axis and the yellow dashed line marks the PNA.

Finally, note that this simulator computes "pure section properties" only, and does not fully describe the behavior of a real structure. Real T-beams suffer from local flange buckling, web shear buckling, lateral-torsional buckling, the non-uniform stress distribution due to shear lag in wide flanges, concrete cracking, and many other phenomena that the section properties cannot capture. Treat this tool as a learning and preliminary-study aid, and run the detailed checks defined in each country's structural codes for actual design.

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