📚 Case Study Practical Exercise: Failure Analysis of a Cantilever Beam | 案例分析实战演练：悬臂梁失效分析
In this guided case study, a critical cantilever beam on a factory conveyor system has fractured repeatedly during operation. You will step through the engineering investigation process required for the WJEC Pre-U Engineering qualification, applying solid mechanics principles, material science, failure analysis and design optimisation. The exercise mirrors real industrial problem-solving and strengthens your ability to evaluate evidence, perform calculations, and justify engineering decisions.
在这个引导式案例分析中，某工厂传送系统的一根关键悬臂梁在运行中反复断裂。你将沿着WJEC Pre-U工程资格认证所要求的工程调查流程，应用固体力学原理、材料科学、失效分析及设计优化方法。本练习模拟真实的工业问题解决过程，能强化你评估证据、执行计算和论证工程决策的能力。
1. Case Background and Problem Definition | 案例背景与问题定义
A 1.2 m long cantilever beam made of hot-rolled mild steel supports a conveyor roller that carries packaged goods. The beam has fractured three times in the past two months at approximately the same location, 200 mm from the fixed support. Each failure caused over 6 hours of production downtime. Management suspects the beam was overloaded, but the operators insist the maximum weight has never exceeded the 5 kN design specification. Your task is to identify the root cause and propose a reliable redesign.
一根长1.2 m的悬臂梁由热轧低碳钢制成，支撑着运送包装货物的传送带托辊。该梁在过去两个月内断裂了三次，断裂位置大致相同，均位于距固定支撑200 mm处。每次故障导致超过6小时的生产停机。管理层怀疑梁超载了，但操作人员坚称最大重量从未超过5 kN的设计规格。你的任务是找出根本原因并提出可靠的重设计方案。
Key data provided: beam cross-section is rectangular, nominal dimensions 50 mm wide × 80 mm deep. The load is applied vertically at the free end via a roller hanger. Visual inspection shows a jagged fracture surface with beach marks and a noticeable reduction in section width near the failure, from 50 mm to 42 mm due to corrosion and fretting. The beam has welded attachments for a guard that were added after initial installation.
提供的关键数据：梁截面为矩形，名义尺寸宽50 mm × 高80 mm。载荷通过辊筒吊架垂直施加在自由端。外观检查显示断裂面呈锯齿状，有海滩纹样，且断裂附近截面宽度明显减小——因腐蚀和微动磨损从50 mm减至42 mm。该梁在初始安装后加装了防护罩焊接附件。
2. Initial Data Collection and Measurements | 初始数据收集与测量
Accurate data collection is the first step in any engineering investigation. The exact geometry at the failure plane was measured: depth remained 80 mm but the effective width was only 42 mm. The distance from the fixed end to the failure plane, L1 = 200 mm; total span to load point, L = 1200 mm. The weight scale monitoring records showed an actual maximum dynamic load of 5.2 kN due to occasional impact when boxes drop onto the roller.
精确的数据收集是任何工程调查的第一步。对断裂平面的几何尺寸进行了测量：高度仍为80 mm，但有效宽度仅为42 mm。从固定端到断裂平面的距离L1 = 200 mm；总跨度到载荷作用点 L = 1200 mm。重量监测记录显示，由于箱子偶尔掉落到辊筒上产生冲击，实际最大动载荷为5.2 kN。
Material certification indicated the beam is structural steel grade S275JR with a minimum yield strength ReH = 275 MPa. Hardness tests on a sample taken from the unfailed region confirmed the material meets specification. Surface roughness and corrosion pits were measured using a profilometer; the worst pit depth was 0.6 mm, introducing a local stress raiser.
材料证书表明该梁为结构钢S275JR，最小屈服强度ReH = 275 MPa。从未失效区域取样的硬度测试确认材料符合规格。使用轮廓仪测量了表面粗糙度和腐蚀坑；最深的坑深为0.6 mm，形成了局部应力集中源。
3. Load and Force Analysis | 载荷与受力分析
Assuming the beam behaves as a linear elastic cantilever, the vertical load at the free end produces both shear force and bending moment along the span. For the failure plane at 200 mm from the support, the lever arm from the load is (1200 – 200) = 1000 mm. The design load was static 5.0 kN, but the dynamic amplification factor from impact was estimated at 1.1, giving a design bending moment M = 5.2 × 10³ N × 1.0 m = 5200 N·m at that section.
假设梁表现为线弹性悬臂梁，自由端的竖向载荷沿跨度同时产生剪力和弯矩。对于距支撑200 mm的失效平面，载荷力臂为 (1200 – 200) = 1000 mm。设计载荷为静载5.0 kN，但根据冲击估算的动载放大系数为1.1，因此该截面的设计弯矩 M = 5.2 × 10³ N × 1.0 m = 5200 N·m。
In case study methodology, it is essential to check whether the loads were purely static or included fatigue cycles. The conveyor operates 16 hours per day, carrying 1200 boxes per hour. Each box passage generates one stress cycle. In 2 months (≈ 60 days), that accumulates: 60 × 16 × 1200 ≈ 1.15 × 10⁶ cycles.
在案例分析方法中，必须检查载荷是纯静态还是包含疲劳循环。该传送带每天运行16小时，每小时运送1200个箱子。每次箱子通过产生一次应力循环。在2个月（≈ 60天）内，累计循环次数为：60 × 16 × 1200 ≈ 1.15 × 10⁶ 次。
4. Bending Moment and Stress Calculation | 弯矩与应力计算
First, compute the section properties at the failure zone using the reduced width. For a rectangular section, elastic section modulus Z = (b × h²)/6. Here b = 42 mm and h = 80 mm.
首先，使用减小后的宽度计算失效区域的截面属性。对于矩形截面，弹性截面模量 Z = (b × h²)/6。此处 b = 42 mm, h = 80 mm。
Z = (42 × 80²) / 6 = (42 × 6400) / 6 = 268800 / 6 = 44 800 mm³ = 4.48 × 10⁻⁵ m³
The bending stress at the extreme fibre is given by σ = M / Z. Substituting M = 5200 N·m:
最外层纤维的弯曲应力为 σ = M / Z。代入 M = 5200 N·m：
σ = 5200 / (4.48 × 10⁻⁵) = 116.07 × 10⁶ Pa ≈ 116 MPa
Compare this to the original design nominal stress if the beam had its full 50 mm width: Z_full = (50 × 80²)/6 = (50 × 6400)/6 = 53 333 mm³, σ_nominal = 5200 / (5.333 × 10⁻⁵) ≈ 97.5 MPa. The corrosion loss increased bending stress by about 19%.
与梁保持完整宽度50 mm时的原始设计名义应力对比：Z_full = (50 × 80²)/6 = 53 333 mm³，σ_nominal = 5200 / (5.333 × 10⁻⁵) ≈ 97.5 MPa。腐蚀损耗使弯曲应力增加了约19%。
5. Material Properties and Allowable Stress | 材料属性与许用应力
For S275JR steel, the minimum yield strength is 275 MPa. Typically, a safety factor of 1.5 on yield is applied for static lifting applications, giving an allowable static stress of about 183 MPa. Even the reduced section stress of 116 MPa is below this, so by a simple static check the beam should not have failed. This indicates that a different mechanism is responsible.
对于S275JR钢材，最小屈服强度为275 MPa。通常，静态起重应用中对屈服取安全系数1.5，得许用静应力约为183 MPa。即使减小截面的应力116 MPa也低于该值，因此根据简单的静力校核，梁不应失效。这表明存在不同的失效机理。
The failure occurred near a welded attachment. The heat-affected zone (HAZ) can exhibit reduced toughness and residual tensile stresses. Moreover, the S275JR steel has a fatigue endurance limit of approximately 140 MPa under fully reversed bending, but here the stress ratio is near zero (pulsating). The effective endurance limit must be corrected using factors for surface finish, size, and stress concentration.
失效发生在焊接附件附近。热影响区（HAZ）可能表现出韧性降低和残余拉应力。此外，S275JR钢材在完全对称弯曲下的疲劳耐久极限约为140 MPa，但此处的应力比接近零（脉动循环）。有效耐久极限必须使用表面粗糙度、尺寸和应力集中系数进行修正。
6. Safety Factor Assessment | 安全系数的评估
A layered safety check is vital. The static safety factor against yielding was 275 / 116 ≈ 2.37, which seems adequate. However, when fatigue is considered, the allowed alternating stress amplitude reduces dramatically. If the corrected endurance limit is estimated at 80 MPa, the stress amplitude of 58 MPa (half of 116) would appear safe, but the presence of the stress concentration from the weld and corrosion pits changes the local peak stress to a much higher value.
分层安全检查至关重要。对屈服的静态安全系数为275 / 116 ≈ 2.37，看似充足。然而，当考虑疲劳时，允许的交变应力幅急剧降低。如果修正后的耐久极限估算为80 MPa，则116 MPa半值58 MPa的应力幅看似安全的，但焊接和腐蚀坑引起的应力集中将局部峰值应力提升到了更高值。
Using a fatigue stress concentration factor Kf of 2.2 for the welded attachment, the local peak stress becomes 116 × 2.2 ≈ 255 MPa. This exceeds the yield strength locally and will certainly cause fatigue crack initiation under 1.15 million cycles. The initial safety factor was misleading because the stress analysis did not account for geometric reduction and stress raisers.
对焊接附件采用疲劳应力集中系数Kf = 2.2，局部峰值应力变为116 × 2.2 ≈ 255 MPa。这已局部超过屈服强度，并肯定会在115万次循环下引发疲劳裂纹萌生。初始安全系数有误导性，因为应力分析未考虑几何削弱和应力集中源。
7. Fracture Surface Examination and Failure Mode Identification | 断裂表面检查与失效模式识别
The fracture surface displayed characteristic beach marks that radiate from a corner at the weld toe. The beach marks are typical of fatigue crack propagation, with smooth, rubbed regions from crack growth and a rough final overload zone. Under SEM, striations were visible, confirming fatigue. The origin was at a corrosion pit adjacent to the weld, where multiple cracks coalesced.
断裂表面显示出从焊趾角落向外辐射的特征性海滩纹样。海滩纹样是疲劳裂纹扩展的典型特征，具有因裂纹扩展而平滑且磨光的区域以及粗糙的最终过载区。在扫描电镜下可见疲劳辉纹，确认为疲劳断裂。裂纹起源于焊接附近的一个腐蚀坑，多条裂纹在此合并。
This evidence directly linked the failure to fatigue exacerbated by stress concentration and corrosion. The weld had not been properly dressed, and the lack of corrosion protection accelerated pitting. The case study now clearly leans toward a fatigue failure, not simple overloading.
这一证据将失效直接与因应力集中和腐蚀而加剧的疲劳联系起来。焊接未经过适当修整，且缺乏防腐保护加速了点蚀。该案例现在明确指向疲劳失效，而非简单的过载。
8. Fatigue and Stress Concentration Analysis | 疲劳与应力集中分析
Applying the modified Goodman relation helps estimate the allowable stress amplitude for the given mean stress. The mean bending stress at the critical section is approximately 116 MPa (tensile). Using the corrected endurance limit of 80 MPa and the ultimate tensile strength of 410 MPa for S275JR:
应用修正的Goodman关系有助于估算给定平均应力下的许用应力幅。关键截面的平均弯曲应力约为116 MPa（拉伸）。采用S275JR修正耐久极限80 MPa和抗拉强度410 MPa：
σₐ / σₑ + σₘ / σᵤ = 1
where σₐ = allowable alternating stress amplitude, σₑ = endurance limit, σₘ = mean stress, σᵤ = UTS. Solving gives σₐ = σₑ (1 – σₘ / σᵤ) = 80 × (1 – 116/410) ≈ 80 × 0.717 = 57.4 MPa. The actual nominal alternating stress half-amplitude was 58 MPa, which is marginally above the allowed 57.4 MPa. However, with Kf = 2.2, the true alternating stress at the notch root is far beyond safe limits.
其中σₐ = 允许的交变应力幅，σₑ = 耐久极限，σₘ = 平均应力，σᵤ = 抗拉强度。求解得σₐ = σₑ (1 – σₘ / σᵤ) = 80 × (1 – 116/410) ≈ 80 × 0.717 = 57.4 MPa。实际名义交变应力半幅为58 MPa，这略高于允许的57.4 MPa。然而，在Kf = 2.2的情况下，缺口根部的真实交变应力远超安全限值。
9. Proposed Redesign Solutions | 改进设计方案
Three redesign options were evaluated: (a) replace the beam with a wider section to restore full 50 mm width and add corrosion protection; (b) redesign the connection to eliminate the welded attachment by using a bolted clamp-on guard, thus removing the weld stress concentration; (c) increase the beam depth to 100 mm and machine a smooth profile, relocating the guard support away from the high-moment region.
评估了三种重设计方案：(a) 更换梁，恢复为完整的50 mm宽截面并增加防腐保护；(b) 重新设计连接，采用螺栓固定式防护罩取代焊接附件，从而消除焊接应力集中；(c) 将梁高度增大至100 mm并加工光滑轮廓，将防护罩支撑移出高弯矩区域。
Option (c) was selected as the most robust. Increasing depth significantly raises section modulus. With b = 50 mm and h = 100 mm, Z_new = (50 × 100²)/6 = 83 333 mm³. The bending stress at the same load reduces to σ = 5200 / (8.333 × 10⁻⁵) = 62.4 MPa. Even with a moderate stress concentration Kf = 1.8, the peak stress is 112 MPa, well below the endurance limit. New fatigue life prediction exceeds 10⁷ cycles, giving infinite life.
选项(c) 被选为最稳健的方案。增加高度可显著提高截面模量。b = 50 mm, h = 100 mm 时，Z_new = (50 × 100²)/6 = 83 333 mm³。相同载荷下弯曲应力降至 σ = 5200 / (8.333 × 10⁻⁵) = 62.4 MPa。即使存在中等应力集中系数Kf = 1.8，峰值应力也仅为112 MPa，远低于耐久极限。新的疲劳寿命预测超过10⁷次循环，达到无限寿命。
10. Verification and Re-calculation | 验证与重新计算
To verify the redesign, a static check and fatigue check were repeated. Static safety factor: 275 / 62.4 ≈ 4.4. Fatigue check using the Goodman diagram: with σₘ = 62.4 MPa, σₑ = 80 MPa, σᵤ = 410 MPa, allowable alternating amplitude = 80 × (1 – 62.4/410) = 80 × 0.848 = 67.8 MPa. The actual alternating amplitude half-value is 31.2 MPa, giving a large margin. Furthermore, the weld is eliminated, so the effective Kf drops to 1.3 for the surface finish, making peak alternating stress only 40.6 MPa, fully safe.
为验证重设计，重新进行了静力校核和疲劳校核。静力安全系数：275 / 62.4 ≈ 4.4。使用Goodman图进行疲劳校核：σₘ = 62.4 MPa, σₑ = 80 MPa, σᵤ = 410 MPa 时，允许交变幅 = 80 × (1 – 62.4/410) = 80 × 0.848 = 67.8 MPa。实际交变幅半值为31.2 MPa，具有较大裕量。此外，焊缝已消除，表面粗糙度的有效Kf降至1.3，使峰值交变应力仅为40.6 MPa，完全安全。
A trial Finite Element Analysis (FEA) of the new beam confirmed uniform stress distribution along the top edge with no singularities, verifying the hand calculation. The simplicity of the improved design also reduces manufacturing risks.
对新梁的有限元分析（FEA）试算证实了上缘应力分布均匀且无奇异点，验证了手算结果。改进设计的简洁性也降低了制造风险。
11. Cost and Manufacturing Considerations | 成本与制造考虑
Although increasing beam depth uses about 25% more material, the elimination of weld repairs and unplanned downtime delivers a payback in under 3 months. Machining the beam from a standard 50 mm × 100 mm hot-rolled bar with a milled radius at the support avoids brittle corners. Bolted guard clamps are off-the-shelf items, minimising custom fabrication. The total modification cost per beam is estimated at £85, compared to £12,000 per hour of lost production.
虽然增大梁高会增加约25%材料用量，但消除焊接维修和非计划停机可在3个月内收回成本。使用标准50 mm × 100 mm热轧棒料加工，并在支撑处铣出圆角以避免脆性转角。螺栓固定的防护罩夹是现货产品，最大限度减少定制加工。每根梁的改造费用估计为85英镑，而与每小时12,000英镑的生产损失相比非常划算。
| Option | Stress (MPa) | Fatigue Life (cycles) | Relative Cost |
| Original (corroded) | 116 | ~5×10⁵ | 1.0 |
| Option (a) restore width | 97.5 | ~2×10⁶ | 0.9 |
| Option (c) deepen + no weld | 62.4 | >10⁷ (infinite) | 1.3 |
12. Final Recommendations and Reporting | 最终建议与报告
The engineering report should conclude that the primary cause of failure was high-cycle fatigue originating at a corrosion pit adjacent to a poorly designed welded attachment. Corrosion reduced the effective section, while the stress concentration from the weld notch amplified local stresses into the low-cycle fatigue regime. The immediate recommendation is to replace all four conveyor beams with the deepened, non-welded design and apply a protective coating system (epoxy zinc-rich primer plus polyurethane topcoat).
工程报告应得出结论，失效的主要原因是高周疲劳，起源于设计不当的焊接附件旁边的腐蚀坑。腐蚀减小了有效截面，而焊缝缺口的应力集中将局部应力放大到了低周疲劳范围。立即建议用加深的非焊接设计替换全部四根传送梁，并施加防护涂层体系（环氧富锌底漆加聚氨酯面漆）。
Additionally, a condition monitoring plan was proposed: six-monthly visual inspections of beam surfaces, and a load cell on the support to record peak loads. This case study demonstrates how a systematic engineering approach—combining field observation, materials data, stress analysis, failure mode identification, and cost-benefit logic—leads to a safe, economic solution.
此外还提出了状态监测计划：每半年对梁表面进行外观检查，并在支撑处安装称重传感器记录峰值载荷。本案例研究展示了一套系统工程方法——结合现场观察、材料数据、应力分析、失效模式识别和成本效益逻辑——如何导向安全、经济的解决方案。
Published by TutorHao | Engineering Revision Series | aleveler.com
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