Structural Analysis Report (CalculiX)

1. Conclusions

• In the current design, the fore-and-aft deflection of the tool tip against a drilling force of 1500 N is 0.033 mm (table surface height 0, tool 150, without stand). On a plate on top of the stand, it is 0.031 to 0.039 mm. Against a side force of 200 N, it is 0.026 mm (fore-and-aft) and 0.028 mm (lateral).
• About 61% of the deflection comes from the linear guides. The guide stiffness is assumed to be 200 N/µm; if it is 100, the deflection is 0.049 mm, and if 400, it is 0.025 mm. We will measure the guides on the actual machine to verify this.
• The decisions that were effective, in order of impact, are: making Z into a box, adding ribs to the legs, adding bulkheads to the beam, shortening the legs to use the spindle closer to the beam, extending the ribs to the front face of the beam, using two head brackets, and widening the Y-rail spacing (Section 3).
• We reduced weight by thinning the plates that have little effect. Lightening holes and building the beam out of square tubing are not worth the trouble (Section 4).
• Strength is more than adequate. The maximum stress is 6 MPa, which is 1/40 of the yield point of steel (SS400), 245 MPa. Plate thickness is determined by deflection, not strength.
• What determines hole positioning accuracy is lead screw error rather than deflection (Machining Methods for Accurate Holes).

2. Current Design

Shapes and dimensions are in Machine Shape and Dimensions, and parts are in the Parts List and Dimension Drawings. Here, we list only the parts relevant to stiffness.

When the spindle is pushed upward by the drilling force, the Z box tilts forward, the beam twists, and the base of the legs bends. The areas with large deflection are the Z box and the guides; within the frame itself, they are the beam and the base of the legs.

3. Design Decisions and Their Effectiveness

Starting from the current design, we solved the model with only one decision reverted at a time. Since all other conditions are kept identical, the increase in deflection represents the effectiveness of that decision. Conditions: table surface height 0, tool 150, without stand.

The difference in weight shows whether it increases (+) or decreases (−) when reverted. The ribs, bulkheads, and brackets weigh just a few kg, yet they reduce deflection significantly.

Decisions Made for Weight Reduction

Thinning the plates saved 8.1 kg, and the increase in deflection was held to +3%. That difference was recovered by extending the ribs.

4. Alternatives Not Adopted

These were solved by adding to or replacing parts in the current design. Closer to the bottom left is better (lighter, with less deflection).

In addition to these, in Section 6.1 of the records, we compared and decided against building the frame from channel steel (either too heavy or too soft), and in Section 6.2, using 3 blocks on each side for Z (effective laterally, but risks the table hitting the ribs) and using a φ20 screw for Z (reduces vertical deflection only).

5. Detailed Results for the Current Design

Spindle Height and Guide Stiffness (Tool 150, Without Stand)

The higher the spindle is raised and the closer it is used to the Z blocks, the smaller the deflection. Guide stiffness is an assumed value, so we will measure it on the actual machine.

Machining a plate on the stand

Stand height is 50, with the plate placed on a shim plate. Values are the relative displacement between the tool tip and the plate. t32 was solved with W 31.

Up/down values include the deflection of the stand plate (when the underside is supported at only 12 points). It is expected to be smaller once mounted on a workbench. It doesn't affect through-holes.

Natural frequencies

At surface height 0: 71, 99, 117, 162, and 202 Hz. Avoid spindle speeds close to the tooth pass frequency. For a 2-flute cutter, 71 Hz corresponds to about 2150 rpm.

Stress

Maximum von Mises stress is 6.2 MPa (legs) under a 1500 N drilling thrust, 1.7 MPa under 200 N fore/aft, and 2.6 MPa under 200 N left/right. These are well below the yield point of steel (245 MPa). Corner values change depending on element size and weld shape, so treat them as rough guides.

Effect by component

In the framework, the beam has the largest effect. The heavy base frame has almost no effect on fore/aft displacement.

6. Record of design studies

The studies leading up to the current design. "Current proposal" or "baseline" in each section refers to the geometry at that point. Values were solved with the design as it stood then, so conditions differ from the values in sections 1–5. Only compare values within the same table.

All deformation plots use the same scale factor (3D: 500x, cross section: 800x) and color map (red for 0.10 mm or more). The stepped edges of the ribs are an artifact of the mesh shape, not the real part profile.

6.1 Framework proposals

The Z section is shared; only the framework was changed. The spindle is at the bottom (spindle nose 130, legs 205).

Plan A: Square tubing. Beam 200×150×4.5, legs 150×75×4.5, frame t3.2

Plan B: Beam C200×80 with t4.5 back plate, legs C150, frame C100

Plan C: Beam C200×80 left open, legs C150, frame C100

Plan D: Beam C200×90 with t4.5 back plate, legs C150, frame C100

Plan E: Two C200×80 beams joined, legs C150, frame C100

Plan F: Beam is square tubing, legs C150, frame C100

Plan G: Bent-sheet assembly. Beam 200×150 front t6 others t3.2, legs t3.2

Plan H: Bent-sheet assembly. Beam 200×180 front t4.5 others t3.2

Plan I: Frame only C100, others same as A

6.2 Plans for the Z section

Framework is common to Plan A.

The plan where the Z plate was made t6 (Z4) is not included in the results because the model nodes failed to connect and no solution was obtained.

Plan Z0: Baseline

Plan Z1: Wall depth 86 → 120

Plan Z2: Add t6 front plate to Z1 to make a box

Plan Z3: Thin the plates in Z2. Walls t6, front plate t4.5

Plan Z5: Head support spacing 200 → 280

Plan Z6: Make the base t12 steel

Plan Z7: 3 Z blocks on each side (add below)

Plan Z8: 3 Z blocks on each side (add above)

Plan Z9: Z guide stiffness 400 N/µm

Plan Z10: All guide stiffness 400 N/µm

Plan Z11: Z screw φ20 (70 N/µm)

Plan Z12: Y rail spacing 140 → 170

Plan Z13: Z2 and Z11

Plan Z14: Z2 and Z12

Plan Z15: Z14 and Z11

6.3 Combination Plans

R8 was to check the mesh; the difference between 15 mm and 10 mm is about 4%. Seeing that raising the spindle reduced front-to-back deflection led directly to the next plan.

Plan R1: Square pipe frame with Z improvements

Plan R2: Bent-plate frame with Z improvements

Plan R3: R2 with 3 Z blocks each

Plan R4: R2 with spindle nose at 230

Plan R5: R2 with spindle nose at 330

Plan R6: R2 with 400 N/µm guides

Plan R7: R2 with 100 N/µm guides

Plan R8: R2 with 10 mm mesh

6.4 Shortening the legs to run the spindle close to the beam

For the same machining plane height, shortening the legs lets the spindle machine closer to the Z blocks. This works well because we mostly machine plates and only need a small distance between the beam and the workpiece.

• With 75 mm legs, the ribs become shorter, making thrust deflection worse than with 105 mm legs. We decided to go with 105 mm legs.
• Increasing rib thickness from t9 to t16 adds 1.2 kg and shifts deflection from 0.038 → 0.036 mm (plane height 0). Not essential.
• Stroke has no effect on deflection when operating at the same position.

6.5 Base height and plate ribs

Front-to-back and left-to-right deflections do not change with base height. What changes is only the plate deflection (up-and-down), which affects blind hole depth.

6.6 Z front plate

Because there is hardly any difference at working positions, we decided not to treat the front plate as a structural member, but rather as a cover. This assumes that the fastening between the head and the bracket is solid.

6.7 Base material

We do not machine aluminum because the chips would get mixed in. We checked whether using steel for the base would also work. Plane height 0, tool 150. A thinner base also brings the spindle closer to the beam.

We decided to use steel t12. Compared to aluminum t25, the tip displacement is only 0.0015 mm larger, it is only 2.3 kg heavier, and there are no vibration issues. The earlier sections (6.1 to 6.6) compared options solved with an aluminum bed, but the relative ranking among the options does not change.

6.8 Lightening the framework (lightening cutouts and plate thicknesses)

We checked whether we could reduce weight while maintaining strength using lightening holes. Calculations were done using the current design (steel t12 bed, surface height 0, tool 150).

Conclusions

• Lightening holes are not worth it. Even if we drill out all the plates where holes can be made (bulkheads, Z-wall, ribs), it only saves 1.7 kg (less than 1% of the machine), while increasing tool tip displacement by up to 10.3%.
• Redistributing plate thicknesses is better. If we drop plates that don't do much by one thickness step and slightly thicken effective areas (front of the legs, ribs), we save 6.2 kg, and displacement actually decreases (see "Light 6" below).
• There is plenty of safety margin for strength (not breaking). The maximum stress is 22 MPa, which is less than 1/10 of the yield point of steel (SS400), 245 MPa. What determines plate thickness is deflection, not strength.

Weight breakdown and effective parts

The framework weighs 80.9 kg, and the rest consists of purchased items like linear guides, the head, motors, and wiring. Only the framework can be changed via cutouts or plate thicknesses.

• What matters most are the guides (linear guides). They are also heavy at 35.7 kg, but if their rigidity drops, displacement increases significantly (0.056 mm in the 6.11 example with 100 N/µm rigidity), so we won't downsize them.
• Within the framework, the beam, foot plates, legs, and ribs are effective. Although the legs and ribs together are light at 3.9 kg, they have a far greater effect than the 6.3 kg bracket.
• The bracket, Z-wall, bulkheads, and frame ends do very little for their weight. These are candidates for weight reduction.

Lightening holes

• Since holes hollow out the middle of a plate, thinning the plate produces less displacement for the same weight reduction. In the bulkheads, saving 0.7 kg with holes increases front-back 200 N displacement by +5.4%, but going from t4.5 to 3.2 saves 1.4 kg with only a +1.1% increase.
• The ribs are small triangles, so holes end up right in the middle of the load path. Even a φ50 hole increases tip front-back displacement by +5.5%.
• Drilling holes in the rear plate of the beam opens up the closed box, weakening it against torsion (+8.6% under front-back 200 N). We will not make holes in the face of the box.
• We will make holes only when there is a specific need, such as routing wiring or piping, venting air during welding, or reaching in to tighten bolts.
• Refining the mesh elements from 15 mm to 7.5 mm showed the same trend (tip front-back: no holes 0.0421, bulkhead holes 0.0432 (+2.6%), rib holes 0.0451 (+7.1%)).

Changing plate thickness by one step

"Worst displacement increase" is the largest increase among four cases: tip fore-aft, tip vertical, 200 N fore-aft, and 200 N left-right. For thickening, the front of the legs and the ribs are the most effective per 1 kg. Thickening the beam or foot plates works, but adds quite a bit of weight.

Weight-reduction options

If you reduce weight purely by plate thickness, we recommend Light 6. If you can extend the ribs to the front face of the beam, Light 7 in Section 6.10 is even better. The frame drops from 80.9 → 74.7 kg (6.2 kg lighter), bringing the entire machine from about 195 → about 189 kg. Tip fore-aft deflection decreases from 0.0397 → 0.0380, vertical from 0.0614 → 0.0602, the 200 N fore-aft and left-right values stay almost the same, and the natural frequency goes up slightly.

The idea is to take weight away from places that don't do much and put just a little back into places that really matter (the front of the legs and the ribs). Thinning the foot plates increases tip deflection, but thickening the legs and ribs makes up for it.

Even when re-solved with a finer 7.5 mm mesh, Light 6 still had less deflection (tip fore-aft: Current 0.0421, Light 6 0.0398, −5.5%. Fore-aft 200 N: Current 0.0293, Light 6 0.0293).

Parts left unchanged: Z plate (thinning to t6 doesn't leave enough thread depth for rail mounting screws), frame long sides (thinning to t2.3 increases left-right deflection by 12% and drops frequency from 70 → 62 Hz), and the beam and base (thinning them hurts too much, so we keep them as is).

Stress

• Peak von Mises stress appears at the top corner of the ribs (where the leg front meets the beam underside) in all options. Corner values vary with mesh size and weld geometry, so take them as rough guides.
• Light 4 and Light 6 thicken the ribs and legs, lowering the maximum stress.
• Stress roughly triples around hole edges, but because the stress in the bulkheads and Z walls is only 1–2 MPa, adding holes causes no strength issues.

6.9 Building the beam from two square tubes (top & bottom) and flat bar

We looked into making the beam by joining two square tubes (top and bottom) with flat bar in between. The outer dimensions match the current design (depth 150, height 200, length 800), with 150×50 square tubes placed top and bottom and the 100 mm gap between them closed with flat bar. Calculations used the current setup (t12 steel base, surface height 0, tool 150).

• We compared three layouts: flat bar covering the full length of both front and back faces, flat bar only on the front, and flat bar bridging five spots (a ladder layout).
• Bulkheads cannot be placed inside the square tubes, so they go between the tubes (behind the flat bar). Both ends are capped with t9 plates.
• Where flat bar overlaps the face of a square tube, it was modeled as a single plate with the combined thickness of the two.

Conclusion

• None of the combinations were lighter and stiffer than our current plate box.
• The stiffest option is adding full-length t6 flat bars to both the front and back, with 3 bulkheads inside. Tip front-back deflection goes from 0.0397 to 0.0371 (−6.5%), but the beam gets 13.6 kg heavier. If we simply increase the front plate of our current box from t6 to t9, it only adds 3.7 kg for a −5.3% reduction in tip front-back deflection (table in 6.8). For the same stiffness, making the current box thicker is lighter.
• The two inner walls of the square tubing (near the middle of the beam height) have little effect on both bending and twisting, so they do not add enough stiffness for their weight. The stiffness of the beam is determined by its 4 outer faces.
• Adding a flat bar only to the front leaves the back open, making it weak against torsion, and the front-back 200 N deflection increases by +15.4%. A ladder structure is even weaker, with tip front-back deflection at +33.5% and up-down at +26.7%.
• Without bulkheads, front-back 200 N deflection increases by 43.2%. If we close it off with flat bars, we must include the bulkheads between them.
• Ease of fabrication is not much different either. Because both edges of the flat bars are welded to the square tubing, there are 4 long weld seams across the front and back, the same as the 4 corner welds on our current box. In every design, the rail mounting surfaces need to be milled flat after welding.

We decided to stick with our current plate box. If we want to make the beam stiffer, we can increase the bulkheads from 3 to 5 (+2.0 kg, front-back 200 N deflection −2.9%), or make the front plate thicker.

Beam weight (kg) is the total of the square tubing or plates, flat bars, bulkheads, and end plates.

6.10 Extending the column ribs to the front face of the beam

The current ribs are triangles connecting the front face of the columns to the foot plates, with their upper ends at the bottom surface of the beam. We looked into extending them upward along the front face of the beam to directly connect the front face of the beam to the foot plates. The front end (on the foot plate, 210 from the front face of the beam) is kept unchanged. Calculations were done with the current design (steel base t12, surface height 0, tool 150, spindle at center of Y).

Conclusions

• Very effective. Extending both sides all the way to the top face of the beam reduces tip front-back deflection from 0.0397 to 0.0305 (−23.2%), with a weight increase of +2.9 kg. Per unit of weight, this is more effective than any of the plate thickness changes investigated in 6.8.
• The front column can only be extended up to 60 mm above the bottom surface of the beam. This is because the Y-axis belt cover sits in front of the beam at the same left-right position (Y 5 to 25). There is nothing in the way of the rear column.
• Practical extended design: front side extended 60 mm up, rear side extended to the top face of the beam. Tip front-back deflection is 0.0320 (−19.4%), up-down is 0.0575 (−6.4%), front-back 200 N is 0.0258 (−7.5%), left-right 200 N is unchanged, and weight is +1.9 kg. Even when refining the mesh to 7.5 mm elements, tip front-back deflection goes from 0.0421 to 0.0338 (−19.7%), showing the same trend.
• Combining this with the weight-reduction options from 6.8 allows us to increase stiffness while cutting weight. By thinning the frame ends, bulkheads, Z-axis walls, brackets, and foot plates while extending the ribs (Light 7), the structural frame becomes 6.2 kg lighter, and tip front-back deflection drops by −17.1%.

Why it works

When drilling thrust pushes the spindle upward, the beam twists as if tipping forward, bending the base of the columns. Because the current ribs only support the base of the columns, bending occurs at the corner of the beam's bottom surface. Extending the ribs up to the front face of the beam provides diagonal support to the front face directly from the foot plates, reducing this bending. This is the same reason the columns and ribs were so effective for their weight in Figure 2 of Section 6.8.

Combining with the lightweighting options from 6.8

The extended-rib shape is "Front 60, back up to top of beam". Light 2 has frame ends t1.6, bulkheads t3.2, Z walls t4.5, and mounts t16.

We recommend Light 7 (replacing Light 6 from 6.8). The frame becomes 6.2 kg lighter, just like Light 6, and the tip front-back displacement is even smaller at 0.0329 compared to 0.0380 for Light 6. The legs and ribs can stay at their current thicknesses (legs: rectangular tube 150×75×3.2, ribs: t9), so there is no need to fabricate the legs out of plate. Only the displacement under left-right 200 N increases slightly by +0.7%.

Notes

• When moving Y all the way to the front (spindle Y 160), the front rib has only about 3.5 mm of clearance to the Y carriage block on the left and right. Either shift the rib 5 mm forward (Y 13) or reduce the Y travel by 5 mm. The back rib has about 27 mm of clearance.
• At the height of the lower rail, the back rib projects forward by about 125 mm from the front face of the beam. It does not collide with the Y carriage or blocks because its left-right position is different, but keep it clear of wiring and cable routing paths.
• Because the rib shapes differ between left and right, the displacement when the spindle is at the ends of the Y axis may vary slightly between left and right. The calculations only cover the case where the spindle is at the center of Y.
• If the Y belt cover can be moved to the opposite end (the back), both sides can be extended to the top of the beam (tip front-back −23.2%). This requires changing the motor and belt layout.

6.11 Results before adopting Light 7 (N1)

Legs 105, Z stroke 150, no Z front plate, steel table t12. Units in mm.

When machining a plate on top of the base stand

Base stand height 50, plate placed on top of a backing plate. t32 was solved using W 31 (W 29 had mesh nodes that did not connect).

About 0.075 of the up-down value is the deflection of the base stand plate, which is the value when supported at only 12 points on the bottom surface. This is expected to be smaller when placed on a workbench. It does not affect through-holes.

Differences by spindle height (tool 150, no base stand)

Natural frequencies

At surface height 0: 71, 90, 108, 153, 198 Hz. Avoid spindle speeds close to the tooth pass frequency. For a 2-flute tool, 72 Hz corresponds to about 2150 rpm.

6.12 Effectiveness of each member (before deciding on the legs)

We investigated this using the baseline design (model 111.5 kg) before deciding on the legs and stroke. The underlying behavior is the same for the current design.

This shows how much the displacement decreases when increasing the stiffness of only that member by 100 times. The guides are the most significant, followed by the beam, then the Z walls and plates. The frame barely has any effect, though this is partly because the supports were assumed to be fixed.

6.13 Which edges to weld on the bulkheads inside the beam

Since the beam is closed by joining the front plate to the channel, the internal bulkheads cannot have all their edges welded from the inside. Using the current design (Sections 1 to 5), we solved for different welded edges on the 3 internal bulkheads. The edges that were not welded were offset 15 away from the beam plate so they would not connect.

• The role of the bulkheads is to maintain the cross-sectional shape of the beam. Connecting the front plate and the back plate is what matters; the top and bottom edges have little effect.
• A bulkhead welded only on the front edge is almost the same as having none at all.
• We decided on an assembly method where the front edge is welded from both sides, and the back edge is welded through slots in the back plate (see "Beam Assembly Method" in Parts List and Dimensional Drawings).

6.14 Lateral Position of the Leg Ribs

We solved with the leg ribs moved to the outer edge, center, and inner edge of the leg square tubing (width 75). Based on the current design (Sections 1 to 5), the far leg has these positions mirrored left-to-right.

• Shifting the rib to the outer edge of the square tubing makes it slightly worse (+3%). Moving it flush with the outer wall gives +5%.
• What matters is whether the rib sits above the X blocks under the foot plate (near Y −5 to 55, far Y 745 to 805). As long as it is over the block, the positional difference is within 2%; once it leaves the block, the foot plate flexes and performance drops significantly (+21% at the inner edge).
• The near rib has only 8.5 of clearance to the Y carriage, so it can barely be moved inward from where it is now. The far side can be moved directly above the block (Y 775), but the effect is only −1%, so we kept the current positions.

6.15 Moving the legs and ribs toward the center of the X blocks

We moved the leg square tubes (width 75) and ribs toward the center of the X blocks under the foot plates (front Y 25, back Y 775). Since the legs stick out 12.5 past the frame, we also extended the beams by 12.5 on both sides to 825, and moved the internal bulkheads to the center of the blocks (Y 25, 775) as well. We solved this with the current configuration (Sections 1 to 5).

• Even if we move the leg square tubes, the deflection barely changes. The only thing that has an effect is the rib position, and even that is just −2%.
• If we place the front rib at Y 25, it hits the block of the Y carriage when Y is moved all the way forward. We would need to reduce the Y travel range by about 12 at the front.
• The beams stick out past the frame, increasing the overall width from 800 to 825.
• It isn't worth changing the travel range and overall width for just 2%, so we decided to keep the current shape as it is.

7. Terminology and Model

Terminology and Coordinates

Model

What Is Not Included

• Actual guide rigidity and play. Welding distortion. Bolt loosening (welds and plate joints are assumed to be fully bonded).
• Spindle bearing deformation, bending of the tool itself, and deformation of the plate and jig.
• Rigidity of the rails and reference stops (we did not add this, so it stays on the safe side).
• Spindle motor position (mass was included in the head). Details of the X nut mount.

8. What We Verified

• Torsion of a single beam: The finite element method average of 5.13×10⁻⁵ rad matched our hand calculations (Bredt's formula) (beam 200×150×4.5, length 800, torsional moment 1500 N × 288 mm).
• Mesh: The difference between 15 mm and 10 mm was about 4–5%. A 15 mm mesh is sufficient for comparisons. We treat absolute values as slightly larger.
• Node connections: We verified that the members are split into four chunks (frame, gantry, table, Z) and connected only by the guide springs.
• Unsolvable cases (Z4, W 29): The grid lines were too close, so the nodes failed to connect. You can tell because the displacement reaches the 10⁸ mm range. We did not use those values.
• Even with the current shape, verifying with 7.5 mm elements showed only about a 5% difference compared to 15 mm (6.10). A 15 mm mesh is enough for comparisons.
• The rear rib collided with the Y ball screw, so we shortened the screw length so it only sticks out 15 beyond the supported-side bearing (BF12) (verified in the 3D model).

9. How to Reproduce

The models and calculation scripts are located in analysis/fea in the repository.