工程数学英语
一、基本运算 | Basic Operations
二、代数与方程 | Algebra & Equations
三、几何与三角 | Geometry & Trigonometry
四、微积分基础 | Calculus Basics
五、统计学与概率 | Statistics & Probability
六、工程应用计算 | Engineering Applied Math
Concrete Mix Design Calculation — A Technical Discussion
Liu, I've been reviewing the concrete mix design. Your target compressive strength is 5,000 psi. Can you walk me through how you arrived at the water-cement ratio?
刘工，我看了你们的混凝土配合比设计。目标抗压强度是5000 psi。你能 给我讲一下水灰比是怎么算出来的吗？
刘工: 当然。We use the ACI 211 method. The target strength f'c is 5,000 psi. We add an over-design margin — standard deviation times 1.34 for the safety factor.
当然。我们采用ACI 211方法。目标强度f'c是5000 psi。我们加上超设 计裕量——标准差乘以1.34作为安全系数。
The required average strength f'cr equals f'c plus 1.34 times the standard deviation. Our historical data shows standard deviation is 500 psi. So: f'cr = 5,000 + 1.34 × 500 = 5,670 psi.
所需平均强度 f'cr = f'c + 1.34 × 标准差。我们历史数据显示标准差 是500 psi。所以：f'cr = 5,000 + 1.34 × 500 = 5,670 psi。
Okay, that makes sense. Then how do you get from f'cr to the water-cement ratio?
好的，说得通。那从f'cr怎么得到水灰比？
刘工: We use the ACI strength versus w/c ratio curve. For 5,670 psi, the curve gives us approximately 0.40 water-cement ratio by weight. But we also have a maximum w/c ratio of 0.45 for durability — so we take the lower value, 0.40.
我们用ACI的强度-w/c比曲线。对于5,670 psi，曲线给出大约0.40的水 灰重量比。但耐久性要求最大水灰比是0.45——所以我们取较小值，0.40。
Good. And how do you calculate the batch weights for a cubic yard?
好。那每立方码的配料重量怎么算？
刘工: Let me walk through the numbers. Our target slump is 3 to 4 inches. From ACI tables, for 1-inch aggregate and 4-inch slump, the mixing water is 325 lbs per cubic yard.
我来算一遍。目标坍落度3到4英寸。查ACI表，1英寸骨料、4英寸坍落度， 用水量是每立方码325磅。
Water = 325 lbs. W/C ratio = 0.40, so cement = 325 / 0.40 = 812.5 lbs per cubic yard. Round up to 815 lbs.
水 = 325磅。水灰比 = 0.40，所以水泥 = 325 / 0.40 = 812.5磅每立 方码。向上取整为815磅。
And the aggregates?
那骨料呢？
刘工: Coarse aggregate: from the table, for 1-inch maximum aggregate with a fineness modulus of 2.8, the coarse aggregate factor is 0.68. Dry-rodded unit weight is 105 lbs per cubic foot. So coarse aggregate = 0.68 × 105 × 27 = 1,928 lbs per cubic yard.
粗骨料：查表，1英寸最大粒径、细度模数2.8，粗骨料系数为0.68。干捣 实容重105磅每立方英尺。所以粗骨料 = 0.68 × 105 × 27 = 1,928磅每立 方码。
Fine aggregate: total volume minus water volume minus cement volume minus coarse aggregate volume, times the specific gravity of sand times 62.4. That gives us about 1,240 lbs per cubic yard.
细骨料：总体积减去水体积减去水泥体积减去粗骨料体积，乘以砂的比重乘 以62.4。得出约1,240磅每立方码。
So the final mix per cubic yard is: cement 815 lbs, water 325 lbs, coarse aggregate 1,928 lbs, fine aggregate 1,240 lbs. Total: 4,308 lbs. Density checks out.
所以每立方码最终配合比是：水泥815磅，水325磅，粗骨料1,928磅，细骨 料1,240磅。合计4,308磅。密度验证通过。
刘工: Correct. That's a 6.5-bag mix — 815 divided by 125 lbs per bag is 6.5 bags of cement per cubic yard.
没错。这是6.5袋配合比——815除以每袋125磅为每立方码6.5袋水泥。
Great. Now I want to convert these to metric for the contractor's batching plant — they're European and use metric.
很好。现在我想把这些换算成公制——承包商的搅拌站是欧洲的，用公制。
刘工: Easy: 1 lb = 0.4536 kg, 1 cubic yard = 0.7646 cubic meter. So per cubic meter: cement = 815 × 0.4536 / 0.7646 ≈ 483 kg/m³.
简单：1磅 = 0.4536公斤，1立方码 = 0.7646立方米。所以每立方米： 水泥 = 815 × 0.4536 / 0.7646 ≈ 483 kg/m³。
Water ≈ 193 kg/m³, coarse aggregate ≈ 1,144 kg/m³, fine aggregate ≈ 736 kg/m³.
水 ≈ 193 kg/m³，粗骨料 ≈ 1,144 kg/m³，细骨料 ≈ 736 kg/m³。
Perfect. I'll verify the trial batch results next week.
完美。我下周验证试配结果。
Progress Tracking — S-Curves and Statistical Analysis
Sun, how are we tracking against the baseline schedule? The monthly report to the client is due Friday.
孙工，我们对照基线进度的执行情况怎么样？周五就要交业主月报了。
孙工: I just updated the S-curve. Overall, we're at 62.3% physical progress versus 65.0% planned. We're lagging by 2.7 percentage points.
我刚更新了S曲线。总的来说，实际进度62.3%，计划65.0%。我们滞后了 2.7个百分点。
That's not great. Which work package is dragging us down?
不太理想。是哪个工作包在拖后腿？
孙工: Civil works is on track — 85% complete. Electrical installation is the problem. Planned was 48%, actual is only 39%. That's a variance of negative 9 percentage points.
土建正常——完成了85%。电气安装是问题所在。计划48%，实际只有39%。 负偏差9个百分点。
If we look at the numbers: planned electrical progress should have been 2,330 out of 4,850 total installation points. Actual completed is 1,892 points. We're short by 438 points.
具体数字：电气计划完成4850个安装点中的2330个。实际完成1892个。 差了438个点。
What's the root cause? Materials? Manpower?
根本原因是什么？材料？人力？
孙工: Both. The MV cable delivery was delayed by three weeks. Plus, the contractor only mobilized 70% of the planned workforce. If I do the math: planned productivity was 12 points per crew per day. With 70% manning, actual is 8.4. That's a 30% drop.
两者都有。中压电缆到货晚了三周。而且承包商只动员了计划70%的人力。 算一下：计划生产率是每组每天12个点。按70%人力，实际只有8.4。下降 了30%。
So if we increase manning to 100% starting next week, when can we catch up?
那如果我们从下周开始把人力拉到100%，什么时候能赶回来？
孙工: Let me run the numbers. We have 4,850 - 1,892 = 2,958 points remaining. At 12 points per crew per day with 6 crews, that's 72 points per day. So: 2,958 / 72 ≈ 41 working days.
我算一下。还剩4850 - 1892 = 2958个点。每组每天12个点，6个组，每 天72个点。所以：2958 / 72 ≈ 41个工作日。
With 6-day work weeks, that's about 7 calendar weeks. The original finish date was in 8 weeks, so we'd still be about 1 week late unless we add overtime or a 7th crew.
按每周6天工，大约7个日历周。原定完成日期是8周后，所以我们还是大约 晚1周，除非加加班或多上一个组。
What if we add Saturday overtime at 150% productivity?
如果周六加班按150%生产率呢？
孙工: Let me recalculate. 6 crews × 12 points × 6 days = 432 per week regular. Plus 6 crews × 12 points × 1.5 × 1 day overtime = 108 per week overtime. Total 540 per week.
重算一下。6组 × 12点 × 6天 = 每周432点常规。加上6组 × 12点 × 1.5 × 1天加班 = 每周108点加班。合计每周540点。
2,958 / 540 = 5.5 weeks. That puts us back on track with a small buffer.
2958 / 540 = 5.5周。这样就能回到正轨了，还有点缓冲。
Good. Now, what about the cost implications? The cost performance index?
好。那成本影响呢？费用绩效指数怎么样？
孙工: CPI is currently 0.94. Earned value is $18.7 million, but actual cost is $19.9 million. We've overspent by $1.2 million.
CPI目前是0.94。挣值是1870万美金，但实际成本是1990万美金。超支了 120万美金。
The variance is mainly from the cable delay — we paid $400K in expediting fees, and idle labor during the delay cost $800K.
偏差主要来自电缆延误——我们付了40万美金催交费，延误期间的窝工损失 80万美金。
Can we claim that back?
这个能索赔回来吗？
孙工: We're working on a time-and-cost claim. The probability of recovery is about 70% based on similar claims we've won. But worst case — if we only recover 50% — our contingency pool still has $2.5 million, so we stay within budget.
我们正在做工期和费用索赔。根据以往类似索赔胜率，追回概率大约70%。 但最坏情况——如果只追回50%——我们应急池还有250万美金，所以预算不会 超。
Good to hear. Put together a recovery schedule with those overtime numbers and include the S-curve forecast. I'll present it at the weekly progress meeting.
不错。把加班的赶工计划和S曲线预测整理出来。我在周进度会上汇报。
孙工: Will do. I'll have it ready by tomorrow noon.
没问题。明天中午前准备好。