Understanding Plate Element Forces in PLAXIS 3D

Units, Interpretation and Practical Use for Geotechnical Engineers

Introduction

In PLAXIS 3D, plate elements represent thin structural components such as slabs, linings, or walls. Their internal forces are reported as stress resultants per unit length: membrane forces (N) in kN/m, shear forces (Q) in kN/m, and bending or twisting moments (M) in kNm/m. Because these values are expressed per metre width, they often appear unfamiliar to users accustomed to beam results in kN or kNm. This article explains the reasoning behind these units, how to interpret them correctly in PLAXIS 3D, and how to convert or integrate them when needed for design and reporting.

Key Takeaways

• Plate results in PLAXIS 3D are reported as stress resultants per unit length:
  • Membrane forces (N) in kN/m
  • Shear forces (Q) in kN/m
  • Bending/twisting moments (M) in kNm/m
• Units reflect plate theory: Plate elements represent thin structures; stresses are integrated across thickness, yielding line resultants.
• Use kNm/m (or kN/m) directly for design: Convert to total kNm (or kN) only when you are checking global equilibrium or load transfer.
• Two‑way plates ≠ independent beam strips: Use integration or design strip methods for total actions; avoid naïve summation.
• Never multiply plate results by mesh element size: Mesh size does not affect unit meaning.
•  Always verify local plate axes and sign conventions before interpreting N₁/N₂, M₁₁/M₂₂/M₁₂, Q₁₂/Q₂₃/Q₁₃. See also: Local axis and its colour indication

Plate Element Forces: Physical Meaning and Units

Plate elements represent thin, two‑dimensional structures with a defined thickness t. In classical plate theory, in‑plane stresses and out‑of‑plane stresses are integrated through the thickness to form stress resultants:

 

Figure 1: Stress resultants in the plate

Why per unit length?

Because the integration removes the thickness dimension, the resulting quantities are distributed per metre along the plate surface, hence the characteristic per-length units.

Interpreting Plate Results in PLAXIS 3D

Checklist for Interpreting Results

• Open the Output program after calculation.
• Select the plate object in the model (via double-click or “Select Structures” tool).
• From the Forces menu, choose the desired force/moment component (N, Q, or M).
• Use the Shadings or Distribution for visual results and Table for numerical data.
• Verify local axes from View → Local Axes to ensure directions match your design interpretations.
• Important: All reported plate forces and moments are per metre width, regardless of mesh size.

Plate vs Beam results: What’s the Difference?

Many engineers are used to beams, where forces and moments are total values in kN or kNm. Plates behave differently.

• Beams are one-dimensional: all stiffness is concentrated along a line, so moments are reported in kNm.
• Plates spread stiffness over an area: results are distributed in kNm/m, meaning moment per metre width.

Worked Example

A cantilever beam under a uniform load (q) (kN/m) and length (L) (m) has a maximum bending moment: 

If the same system is modelled in PLAXIS 3D as a plate 1m wide, PLAXIS reports:

The value is numerically the same, but the unit is “per metre” because the plate could extend beyond 1m width.

For this example, we consider a 10m cantilever beam with a uniform pressure of (q=100 kN/m²). For the beam analogy we use a 1 m strip so (q' = q · 1 = 100 kN/m).

The figure below shows the bending-moment diagram M(x)=q' · (L-x)²/2 (kNm) from the fixed support (x=0) to the free end (x=10 m). The maximum moment at the support M_max = q' · L²/2 = 5,000 kNm is annotated.

Figure 2: Cantilever beam analogy (L = 10m) under uniform pressure

Example 1: Cantilever plate, primarily one-way bending

To understand the unit convention in PLAXIS 3D, it helps to use the analogy of a cantilever beam using a plate element. Consider a cantilever plate of length (L = 10m) and width (w = 1m). The plate is subjected to a uniform vertical surface load of: 

q = 100 kPa = 100 kN/m²

Since the width is 1m, we can convert the surface load into a line load: 

q' = q × 1 m = 100 kN/m

This allows us to treat the plate as a cantilever beam strip of unit width.

Figure 3: Model of a cantilever plate under uniform pressure (w = 1m)

For a cantilever beam under uniform load, the maximum bending moment at the fixed end is given by:

Since this calculation was carried out for a strip of width 1m, the bending moment is expressed per unit width:

M_max = 5,000 kNm/m

Figure 4: Bending moment diagram (w=1m)

Example 2: Same plate, width increased to b = 2 m

Now the total applied surface load is spread over a plate 2 m wide. Therefore, the line load over the entire plate is:

q'_total = q · w = 100 × 2 = 200 kN/m

Figure 5: Bending moment diagram (w=1m)

 

The maximum total bending moment will be:

But PLAXIS reports moments per unit width. Since the plate is 2 m wide, the per-unit-width bending moment is:

 

Figure 6: Bending moment diagram (w=2m)

The reported bending moment per unit width does not change with plate width. If the plate becomes 2m wide instead of 1m, the load is spread over a wider area, but the bending moment per unit width remains the same because the plate still behaves similarly to a cantilever beam. So, PLAXIS always outputs results in kNm/m, consistent with the unit-width assumption of plate elements.

One-Way vs. Two-Way Bending

In reality, how a plate resists load depends on its geometry, boundary conditions, and stiffness ratios. If the plate is long and narrow, or if it is only supported/fixed along one edge (like our cantilever), the load is mainly carried in one direction. The plate essentially behaves like parallel beam strips, each resisting load independently. In this case, the hand calculation analogy we did earlier is exact.
However, if the plate is supported along two (or more) edges, or for very wide plates, the load is distributed in two directions. For example, a rectangular plate fixed on all sides develops bending moments in both longitudinal and transverse directions.

• PLAXIS accounts for this automatically using plate theory.
• The results are still reported as moments per unit width (kNm/m), but now you will see significant moments in both directions (M₁₁ and M₂₂), as well as possible twisting moments (M₁₂).

Thus, the interpretation of units remains the same, but the distribution of bending moments changes depending on whether the plate response is one-way or two-way.

Computing Total Forces and Moments

As we have established, in plate elements, axial and shear forces are given in kN/m and bending moments are given in kNm/m. These are typically the desired units when designing reinforcements for slabs, since reinforcement is normally calculated for a 1m strip width. Also, as we mentioned two-way bending plates (e.g., square plates fixed on all edges) do not behave like beams. However, there are cases where calculating the total force in kN (instead of kN/m) or total moment in kNm (instead of kNm/m) makes sense.
For example, when designing supporting structures (beams, foundations, walls, columns) that carry loads and moments from plates or slabs, when checking global equilibrium of the structure or when validating FEA results with hand calculations in beam theory, it may be required to use total rather than per-width moments.

How to compute total bending moments from PLAXIS results

The total bending moment over a finite strip or the entire plate is obtained by integrating the per-unit-width moment across the relevant width, yielding units kNm (not kNm/m).

A. Simple constant (one-way) case – multiply by width

If you are analysing a narrow plate that behaves like a beam, it makes sense to convert bending moments into kNm to match beam conventions. In those cases, the plate response is one-way, and the per-unit-width moment is constant across the width (w):

M_total = M_{per m} × w

Example (from earlier): M_{per m} = 5,000 kNm/m, w = 2 m → M_total = 5,000 × 2 = 10,000 kNm.

B. Two-way bending (spatially varying moments)

In two-way bending the moment field varies with both in-plane coordinates. To get the total moment about, say, the x-axis for a plate region 0 ≤ y ≤b at a particular x:

and similarly, for M_y. 
Note on M_xy: M_xy is the twisting moment per unit width. It does not directly add algebraically to M_x or M_y totals, but it affects principal moment directions and stresses. If you need principal bending resultants, compute the in-plane moment tensor at each location and find principal values (eigenvalues) before integrating if that is what your design requires.

Discrete (numerical) computation – what you actually do with PLAXIS data

PLAXIS can export nodal/element values of N₁, N₂, Q₁₂, Q₂₃, Q₁₃, M₁₁, M₂₂, M₁₂. To compute totals numerically:

1. Decide the resultants you need (e.g., Sectional bending moment about the local x-axis at several x-cuts: M_x,section (x), Sectional axial resultants N_x ​(x), Sectional shear V_x (x).
2. Create a cross section at the desired location.
3. Export data from PLAXIS as needed by filtering the data directly from the table results of the plate.
4. Choose integration strategy. For example, for a desired cut at x = x₀, build a 1-D sampling of M_x (x₀,y) and numerically integrate across y using the trapezoid rule or Simpson:

Figure 7 shows the graphical representation of the trapezoidal summation (integration) of bending moments across the plate width at x=0:

• The blue curve shows M_x (0,y), the bending moment per unit width (kNm/m).
• The orange trapezoids represent each subarea in the trapezoidal rule - they approximate the total area under the curve.
• The sum of these areas gives the total bending moment across the section:

M_total (x=0) ≈ 131.25 kNm

This total is the integrated bending moment about the fixed edge - exactly how finite element post-processing combines nodal or Gauss point results into global section quantities.

Optional: Perform a mesh sensitivity study. Refine the plate and adjacent soil mesh and confirm that reported N, Q, M fields stabilise.

Figure 7: Trapezoidal summation of bending moments across plate width

Common Pitfalls and How to Avoid Them

• Multiplying by element size: Plate forces/moments are not per element; they are per metre width at the reported location.
• Summing two-way fields as if they were independent beams: for square or nearsquare panels on multiple supports, the moment distribution is twodimensional.
• Ignoring local axes/signs: confirm PLAXIS local axes orientation (in PLAXIS it is local 1-2) and the positive bending sign convention before reading M₁₁/M₂₂.
• Over‑interpreting red spots: stress/resultant singularities occur near point loads or restraints; use averaged values away from singularities and verify mesh refinement.
• Skipping equilibrium checks: integrate reactions along supports and compare to applied loads; mismatches often indicate interpretation or boundary issues.

References and Further Reading

• PLAXIS 3D Reference Manual and Material Models Manual (Bentley Systems).
• Timoshenko, S., Woinowsky‑Krieger, S. (1959). Theory of Plates and Shells. McGraw‑Hill.
• Ugural, A. C. (2018). Stresses in Plates and Shells. CRC Press.
• Zienkiewicz, O. C., Taylor, R. L., Zhu, J. Z. (2013). The Finite Element Method: Its Basis and Fundamentals. Elsevier.