In my recent investigation, I have explored how casting defects affect the mechanical behavior of steel structures that incorporate cast steel joints. Cast steel joints are widely used in large-span buildings, bridges, and offshore platforms because of their smooth force transfer and flexible geometric shaping. However, during the manufacturing process, internal discontinuities such as shrinkage porosity, gas pores, slag inclusions, and micro-cracks inevitably appear inside the castings. These casting defects may significantly alter the local stress fields and reduce fatigue life, even though the global static response may appear unchanged. My work aims to quantify the influence of casting defect size, position, and distribution on the static strength, stiffness, and fatigue performance of a steel truss structure containing a K-type cast steel joint.
The truss model in my study represents a typical steel truss bridge segment. I created a full three-dimensional solid model using SolidWorks and then imported it into ANSYS for finite element analysis. The upper chord was an I-section steel, the lower chord was a circular steel tube, and the diagonal braces were connected through a cast steel node. A concentrated load was applied at the midspan of the upper chord. This configuration is representative of many real bridge and roof structures where cast steel joints are used to connect tubular members.
1. Finite Element Modeling of Casting Defects
Because cast steel joints have complex curved surfaces, modeling casting defects directly in ANSYS is difficult. I therefore used SolidWorks to build the defect geometry and the entire assembled truss. The casting defects were modeled as spherical concave pores, which are representative of gas porosity in steel castings. According to the standard for ultrasonic testing and quality grading of steel castings, quality grades range from 1 to 5, with grade 1 representing the best quality. I selected grade 2 and grade 3 defect dimensions for my parametric study. A grade 2 defect has a thickness dimension of 3 mm and a surface radius of 16.1 mm, whereas a grade 3 defect has a thickness of 4 mm and a surface radius of 35.0 mm.
Defect locations are not deterministic in practice. Based on previous investigations of full-scale cast steel nodes, I identified nine representative positions, labeled ① through ⑨, where casting defects are most likely to appear. These positions include the brace-to-chord intersections, the crown points, and the saddle regions of the joint. I then created multiple finite element models with different combinations of defect positions and grades. The models can be grouped into four categories:
- Single grade 2 defect at each of the nine positions (nine models).
- Single grade 3 defect at five selected positions (five models).
- Multiple grade 2 defects at three randomly selected positions (four models).
- Multiple grade 2 defects at five positions simultaneously (one model).
The finite element mesh was generated using 10-node tetrahedral solid elements (Solid 187). To balance accuracy and computational cost, I performed a mesh sensitivity analysis. I selected a model with a grade 2 defect at position ② and refined the mesh gradually around the defect. Table 1 lists the four mesh schemes considered in the sensitivity study.
| Region | Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 |
|---|---|---|---|---|
| I-section beam | 50 | 79 | 79 | 100 |
| Steel tubes | 15 | 20 | 30 | 40 |
| Cast steel joint | 10 | 15 | 20 | 30 |
| Defect vicinity | 1 | 2 | 4 | 5 |
Under identical loading conditions, the first three mesh schemes produced nearly identical stress and displacement results, while the fourth scheme gave noticeably different values. Therefore, I adopted the third mesh scheme for all subsequent computations to guarantee both accuracy and efficiency.
2. Static Performance of the Structure with Casting Defects
2.1 Influence of Single Defect Position on Static Strength
To study the effect of defect position on static strength, I built nine models, each containing a single grade 2 defect at one of the nine locations. A concentrated load of 1000 kN was applied downward at the midspan of the upper chord. The maximum von Mises stress in the cast steel joint was extracted as the strength indicator. The global stress distribution of the entire truss was almost unaffected by the presence of a single defect. Figure 3(a) in the original study showed the global stress contour when the defect was at position ②, but here I focus on quantitative comparisons.
Table 2 summarizes the maximum von Mises stress for each defect position. For the flawless joint, the maximum stress was 180 MPa. When a grade 2 defect was introduced at different positions, the maximum stress varied from 180 MPa to 201 MPa. The largest difference between different defect positions was 11.7%. This indicates that the position of a single casting defect can moderately alter the peak stress in the joint, but the change is not severe enough to threaten static strength because all values remained well below the yield strength of GS-20Mn5V cast steel (280 MPa).
| Defect position | Maximum von Mises stress (MPa) | Stress concentration factor relative to flawless joint |
|---|---|---|
| ① | 200 | 1.111 |
| ② | 182 | 1.011 |
| ③ | 182 | 1.011 |
| ④ | 180 | 1.000 |
| ⑤ | 180 | 1.000 |
| ⑥ | 180 | 1.000 |
| ⑦ | 180 | 1.000 |
| ⑧ | 180 | 1.000 |
| ⑨ | 200 | 1.111 |
Interestingly, the maximum stress in the joint did not always occur at the defect itself. In some models, the peak stress appeared at the lower-left area of the main chord because of the global bending action of the truss. The defect only caused a local stress concentration in its immediate vicinity. This finding suggests that a small casting defect may not govern the static strength of the entire joint unless it is located in a region of high nominal stress.
2.2 Influence of Single Defect Position on Static Stiffness
I also extracted the displacement distributions for all nine models. The overall vertical displacement of the truss was largest at the midspan of the upper chord, and it decreased gradually toward the supports. The displacement contour of the defective models was practically identical to that of the flawless model. The maximum displacement difference between any two models was less than 0.1%. The local displacement distribution in the cast steel joint also showed no visible variation when the defect position changed. Thus, the presence of a single small casting defect has a negligible effect on the global and local stiffness of a steel truss structure.
2.3 Influence of Defect Size on Static Strength and Stiffness
Next, I examined the effect of defect grade while keeping the position fixed. I introduced a grade 3 defect at positions ①, ②, ③, ④, and ⑨, and compared the resulting maximum stresses with those from the grade 2 models. The comparison is shown in Table 3.
| Model | Defect position | Defect grade | Maximum stress (MPa) |
|---|---|---|---|
| 1 | ① | Grade 2 | 200 |
| 2 | ① | Grade 3 | 202 |
| 3 | ② | Grade 2 | 182 |
| 4 | ② | Grade 3 | 182 |
| 5 | ③ | Grade 2 | 182 |
| 6 | ③ | Grade 3 | 182 |
| 7 | ④ | Grade 2 | 180 |
| 8 | ④ | Grade 3 | 183 |
| 9 | ⑨ | Grade 2 | 200 |
| 10 | ⑨ | Grade 3 | 200 |
The maximum stress difference between grade 2 and grade 3 defects at the same position was only 1.7%. This demonstrates that increasing the defect size from grade 2 to grade 3 has a very small influence on the static strength of the joint. Furthermore, the displacement fields of the grade 3 models were almost indistinguishable from those of the grade 2 models. Therefore, defect size does not noticeably affect the static stiffness of the structure.
2.4 Influence of Multiple Casting Defects on Static Performance
Real cast steel joints may contain multiple defects located at different positions. I created five additional models with combinations of defects, as listed in Table 4. The maximum von Mises stress in each model was extracted under the same 1000 kN concentrated load.
| Model | Defect positions | Defect grades | Maximum stress (MPa) |
|---|---|---|---|
| 11 | ①, ②, ③ | All grade 2 | 182 |
| 12 | ①, ③, ④ | All grade 2 | 198 |
| 13 | ②, ⑦, ⑨ | All grade 2 | 182 |
| 14 | ④, ⑤, ⑥ | All grade 2 | 183 |
| 15 | ①, ②, ③, ⑧, ⑨ | All grade 2 | 186 |
All models remained elastic, since the maximum stress was below 280 MPa. The overall stress distributions were similar to the single-defect models. However, the distribution of defects changed the peak stress to some extent. For example, models 11, 12, and 15 all contained a defect at position ①, yet their maximum stresses were 182 MPa, 198 MPa, and 186 MPa, respectively. These values differ from the 200 MPa obtained for the single-defect model at position ①. This indicates that the interaction between multiple casting defects can either increase or decrease the stress concentration at a given defect site, depending on the relative locations.
From a quality grading perspective, all five models in Table 4 would be classified as grade 2 according to the conventional standard because the most severe defect in each model is grade 2. However, their actual stress responses differ. If one ranks the models by maximum stress, the order is not the same as the quality grade ranking. This disparity reveals the inadequacy of evaluating cast steel joint quality solely by the size of the worst defect.
3. Fatigue Performance of the Structure with Casting Defects
While static performance is important, fatigue is often the governing failure mode for structures subjected to repeated loads. Casting defects act as initial discontinuities that strongly affect crack initiation life. To investigate this, I applied a constant-amplitude fatigue load at the midspan of the upper chord. The maximum load was \(f_{\text{max}} = 1000 \, \text{kN}\) and the minimum load was \(f_{\text{min}} = -1000 \, \text{kN}\), giving a fully reversed loading cycle. The fatigue life of each model was estimated using a modified stress-life (S-N) curve for the cast steel material GS-20Mn5V.
3.1 Modified S-N Curve for Cast Steel with Defects
I used the lower-bound S-N curve obtained from previous fatigue tests on smooth specimens of GS-20Mn5V. The lower-bound curve is expressed as:
\[
\lg N = 28.9028 – 9.9581 \cdot \lg S
\]
where \(N\) is the number of cycles to failure and \(S\) is the nominal stress amplitude in MPa. Because the actual cast steel joint has geometric discontinuities and surface roughness, I introduced a comprehensive correction factor \(K_{\sigma D}\) to modify the smooth-specimen curve. The modified curve becomes:
\[
\lg N = 28.9028 – 9.9581 \cdot \lg \left( S K_{\sigma D} \right)
\]
The corresponding modified fatigue limit is:
\[
\sigma_{-1D} = \frac{\sigma_{-1}}{K_{\sigma D}}
\]
where \(\sigma_{-1}\) is the fatigue limit of the smooth specimen. The correction factor is computed as:
\[
K_{\sigma D} = \frac{K_{\sigma s}}{\varepsilon \beta_1}
\]
In my analysis, \(K_{\sigma s}\) was taken as unity because the stress extracted from the finite element model already included the local stress concentration effect of the defect. The size coefficient \(\varepsilon\) depends on the wall thickness at the defect location, and the surface machining coefficient \(\beta_1\) was taken as 0.65. Table 5 lists the size coefficients for the nine defect positions considered in this study.
| Defect position | Size coefficient \(\varepsilon\) |
|---|---|
| ①, ⑨ | 0.95 |
| ②, ④, ⑧ | 0.90 |
| ③, ⑤ | 0.83 |
| ⑥ | 0.82 |
| ⑦ | 0.80 |
Using these coefficients, I obtained the modified S-N curves for each defect position. The curves are shifted downward for positions with smaller size coefficients, meaning that defects at those positions produce shorter fatigue lives for the same nominal stress amplitude. The fatigue life of a defective joint is then determined by substituting the local stress amplitude at the defect into the corresponding modified S-N curve.
3.2 Effect of Single Defect Position on Fatigue Life
I first examined the fatigue behavior of nine models, each containing a single grade 2 defect at one of the nine positions. The local maximum stress amplitude at each defect was extracted, and the fatigue life was obtained from the modified S-N curves. Table 6 summarizes the results.
| Model | Defect position | Fatigue life (cycles) |
|---|---|---|
| A | ① | \(2.43 \times 10^5\) |
| B | ② | \(3.49 \times 10^5\) |
| C | ③ | \(1.91 \times 10^5\) |
| D | ④ | Infinite (stress below fatigue limit) |
| E | ⑤ | Infinite (stress below fatigue limit) |
| F | ⑥ | Infinite (stress below fatigue limit) |
| G | ⑦ | Infinite (stress below fatigue limit) |
| H | ⑧ | \(4.94 \times 10^6\) |
| I | ⑨ | \(2.365 \times 10^6\) |
According to the standard quality grading system, all nine models have the same quality grade (grade 2). However, their fatigue lives differ by more than two orders of magnitude. Defects at positions ①, ②, and ③ produce relatively short fatigue lives, ranging from \(1.91 \times 10^5\) to \(3.49 \times 10^5\) cycles. In contrast, defects at positions ④, ⑤, ⑥, and ⑦ result in infinite life because the local stress amplitude is below the modified fatigue limit. This dramatic difference demonstrates that the position of a casting defect is far more important than its size when fatigue is the governing design criterion. Positions ①, ②, and ③ should be classified as high-risk zones for fatigue crack initiation.
The fatigue life ratio between the most critical and least critical defect positions can be expressed as:
\[
\frac{N_{\text{max}}}{N_{\text{min}}} = \frac{\infty}{1.91 \times 10^5} \to \infty
\]
Even if we exclude the infinite-life cases, the finite lives range from \(1.91 \times 10^5\) to \(4.94 \times 10^6\), which is a factor of about 25.8. If we compare the high-risk positions ①, ②, and ③ with the lower-risk position ⑧, the ratio is approximately 25.9. This huge scatter in fatigue life cannot be captured by a quality grade that only considers defect size.
3.3 Effect of Defect Size on Fatigue Life
To evaluate the influence of defect grade, I modeled grade 3 defects at positions ①, ③, ④, and ⑨. Table 7 compares the fatigue lives with those of the corresponding grade 2 models.
| Model | Defect position | Defect grade | Fatigue life (cycles) |
|---|---|---|---|
| J | ① | Grade 3 | \(0.55 \times 10^5\) |
| K | ③ | Grade 3 | \(0.68 \times 10^5\) |
| L | ④ | Grade 3 | Infinite |
| M | ⑨ | Grade 3 | \(2.90 \times 10^5\) |
Comparing model J (\(0.55 \times 10^5\)) with model A (\(2.43 \times 10^5\)) shows that increasing the defect grade from 2 to 3 reduces the fatigue life by a factor of 4.4 at position ①. Similarly, at position ③, the fatigue life drops from \(1.91 \times 10^5\) to \(0.68 \times 10^5\), a reduction factor of 2.8. At position ⑨, the reduction is from \(2.365 \times 10^6\) to \(2.90 \times 10^5\), a factor of 8.2. Thus, the influence of defect size on fatigue life is substantial, although the same size change had only a minor effect on static strength. The fatigue life difference between grade 2 and grade 3 defects at the same position is roughly one order of magnitude in some cases.
This observation is consistent with the fatigue crack initiation mechanism. A larger casting defect creates a higher local stress concentration and a larger initial crack-like discontinuity, which accelerates the crack initiation and early propagation process. The relationship between fatigue life reduction and defect size can be approximated by a power law:
\[
N \propto a^{-m}
\]
where \(a\) is the characteristic defect dimension and \(m\) is an exponent between 1 and 3 depending on the material and stress state. In my models, changing the defect radius from 16.1 mm to 35.0 mm while also increasing the through-thickness dimension from 3 mm to 4 mm resulted in a reduction of fatigue life by a factor of 2.8 to 8.2, which implies an effective exponent on the order of:
\[
m \approx \frac{\lg(N_2/N_1)}{\lg(a_1/a_2)}
\]
3.4 Effect of Multiple Casting Defects on Fatigue Life
In actual castings, defects are rarely isolated. I therefore investigated four models with multiple grade 2 defects. The fatigue life was evaluated at each defect location, and the smallest fatigue life among all defect locations was taken as the fatigue life of the joint. Table 8 presents the results.
| Model | Defect positions | Fatigue life at each defect (cycles) | Minimum life (cycles) |
|---|---|---|---|
| N | ①, ②, ③ | ①: \(0.34 \times 10^5\); ②: \(1.64 \times 10^5\); ③: \(1.64 \times 10^5\) | \(0.34 \times 10^5\) |
| O | ①, ③, ④ | ①: \(0.32 \times 10^5\); ③: \(1.64 \times 10^5\); ④: Infinite | \(0.32 \times 10^5\) |
| P | ②, ⑧, ⑨ | ②: \(1.52 \times 10^5\); ⑧: Infinite; ⑨: \(8.67 \times 10^5\) | \(1.52 \times 10^5\) |
| Q | ①, ②, ③, ⑧, ⑨ | ①: \(0.36 \times 10^5\); ②: \(1.64 \times 10^5\); ③: \(1.64 \times 10^5\); ⑧: Infinite; ⑨: \(8.67 \times 10^5\) | \(0.36 \times 10^5\) |
In models N, O, and Q, the defect at position ① gave a fatigue life of about \(0.32 \times 10^5\) to \(0.36 \times 10^5\) cycles, which is more than six times shorter than the \(2.43 \times 10^5\) cycles obtained when the same defect at position ① was the only defect in the joint. This indicates that the presence of other casting defects can further reduce the fatigue life at a given location, even though the local stress amplitude at that location might not change dramatically. The interaction effect may be due to a redistribution of load paths or a change in the global stress field around the joint.
Similarly, for position ②, models N, P, and Q gave fatigue lives of \(1.64 \times 10^5\), \(1.52 \times 10^5\), and \(1.64 \times 10^5\), respectively, all lower than the \(3.49 \times 10^5\) cycles of the single-defect model. Therefore, the fatigue performance of a joint with multiple casting defects is always worse than a joint containing only one defect, even when the additional defects are located far from the critical position.
From a design perspective, this means that if a cast steel joint contains multiple quality-grade-2 defects, the fatigue life cannot be conservatively assessed by considering only the most severe defect. The combined effect of multiple defects must be taken into account.
4. Discussion on Casting Quality Assessment
The traditional quality assessment method for steel castings, as given in the relevant standard, places a reference frame of 317 mm × 317 mm on the region with the most severe defects and distinguishes five quality grades based on the defect size inside the frame. This method implicitly assumes that the largest defect is the most critical for the mechanical performance of the casting. My numerical results challenge this assumption.
To illustrate the inconsistency, I compared the quality grades and the actual static stress rankings for several models. According to the standard, all models with grade 2 defects are considered equally qualified. However, Table 9 compares the quality grade and the actual mechanical ranking based on maximum stress for selected models.
| Model | Defects | Conventional grade | Maximum stress (MPa) | Stress-based rank |
|---|---|---|---|---|
| 4 | ②, Grade 3 | Grade 3 | 182 | 1 |
| 6 | ③, Grade 3 | Grade 3 | 182 | 1 |
| 11 | ①, ②, ③, Grade 2 | Grade 2 | 182 | 1 |
| 13 | ②, ⑦, ⑨, Grade 2 | Grade 2 | 182 | 1 |
| 14 | ④, ⑤, ⑥, Grade 2 | Grade 2 | 183 | 4 |
| 15 | ①, ②, ③, ⑧, ⑨, Grade 2 | Grade 2 | 186 | 5 |
| 12 | ①, ③, ④, Grade 2 | Grade 2 | 198 | 6 |
| 2 | ①, Grade 3 | Grade 3 | 202 | 7 |
It is evident that a model with grade 3 defects (models 4 and 6) can have a lower maximum stress than several models with grade 2 defects (models 12, 15). Conversely, a grade 2 model with defects at certain positions can be more critical than a grade 3 model. This discrepancy is even more pronounced in fatigue life. For instance, a grade 2 defect at position ⑧ leads to \(4.94 \times 10^6\) cycles, while a grade 3 defect at position ③ gives only \(0.68 \times 10^5\) cycles. Both models would be assigned different quality grades under the standard, but the fatigue life ratio is about 72.7 in favor of the higher-grade specimen. If we compare two grade 2 models, one with a defect at position ① and another with a defect at position ④, the former has a finite life of \(2.43 \times 10^5\) cycles while the latter has infinite life. Such a huge difference is completely hidden by the conventional quality grade.
Therefore, I conclude that a rational casting quality assessment method should incorporate not only the defect size but also the location and spatial distribution of defects. A possible approach is to construct a risk matrix that combines the defect severity and the local stress demand. For each potential defect location, a permissible defect size can be back-calculated from the fatigue life requirement. This would result in a location-dependent quality acceptance criterion, which is more consistent with the damage tolerance philosophy.
The total stress concentration factor at a defect can be expressed as a product of geometric and material factors:
\[
K_t = 1 + 2 \sqrt{\frac{a}{\rho}} g\left(\frac{a}{t}, \theta\right)
\]
where \(a\) is the defect depth, \(\rho\) is the root radius, \(t\) is the local wall thickness, and \(\theta\) is the orientation angle. For the same defect size, a larger \(a/t\) ratio or a smaller \(\rho\) leads to a higher stress concentration. This explains why the same grade of defect can have very different effects at different positions.
In fatigue assessment, one can estimate the fatigue life using a fracture-mechanics-based integration:
\[
\frac{da}{dN} = C (\Delta K)^m
\]
where \(C\) and \(m\) are material constants, and \(\Delta K\) is the stress intensity factor range. The initial crack length \(a_0\) is usually taken as the defect size. For a surface defect under remote stress amplitude \(\Delta S\), the stress intensity factor is:
\[
\Delta K = Y \Delta S \sqrt{\pi a}
\]
where \(Y\) is a geometry factor that depends on the defect location and the joint shape. Integration of the crack growth equation from the initial defect size to the critical crack size yields the fatigue life:
\[
N = \int_{a_0}^{a_c} \frac{da}{C \left[ Y \Delta S \sqrt{\pi a} \right]^m}
\]
This equation clearly shows that both the initial defect size \(a_0\) and the geometry factor \(Y\) (which is position-dependent) have a strong influence on \(N\). My finite element results are consistent with this theoretical framework.
5. Concluding Remarks
Through extensive finite element simulations of a steel truss structure containing a cast steel joint with various casting defects, I have drawn the following conclusions:
- Casting defects have a negligible effect on the static stiffness of the truss structure. Both the overall and local displacement distributions remain almost unchanged regardless of defect size, position, or distribution.
- The effect of casting defects on static strength is also small. The maximum stress varies by at most 11.7% when the position of a single grade 2 defect changes, and by only 1.7% when the defect grade changes from 2 to 3 at the same position. All stress values remain below the yield strength, so the static safety of the structure is not controlled by these small defects.
- In contrast, casting defects have a dramatic effect on fatigue performance. A single grade 2 defect can lead to fatigue lives ranging from \(1.91 \times 10^5\) cycles to infinite life depending on its position. This scatter spans more than two orders of magnitude.
- Increasing the defect grade from 2 to 3 reduces the fatigue life by a factor of roughly 3 to 8 at the positions studied, which is equivalent to a one-order-of-magnitude reduction in life.
- When multiple casting defects coexist in a joint, the fatigue life at a given critical position is further reduced compared with a single-defect model. The fatigue performance of a multi-defect model is always inferior to that of a model containing only one defect.
- The conventional method of evaluating cast steel quality solely by the most severe defect size is insufficient. A more reliable assessment should consider defect location, distribution, and the local stress demand, especially for fatigue-loaded structures.
My study highlights the importance of casting defects in the fatigue design of steel structures with cast steel nodes. For engineering practice, I recommend that critical regions of cast steel joints, such as positions ①, ②, and ③ in this study, receive stricter quality control during manufacturing. Nondestructive inspection should be used not only to identify the largest defect but also to map the spatial distribution of defects. Quantitative acceptance criteria based on fatigue life should be established for each identified defect-prone zone.
Future work should incorporate probabilistic distributions of defect size and location to provide a more realistic assessment of structural reliability. In addition, experimental verification of the fatigue life predictions will be valuable. Nevertheless, the current numerical results provide clear evidence that casting defects cannot be ignored in the fatigue design of cast steel joint structures.