[{"data":1,"prerenderedAt":62},["ShallowReactive",2],{"q-106180-arch_structure-9":3},{"question":4,"neighbors":27,"sameNumber":46,"prev":35,"next":38},{"id":5,"classKey":6,"session":7,"year":8,"subject":9,"number":10,"groupStem":11,"ownStem":12,"stem":12,"options":13,"answer":18,"answerKind":20,"answerS":11,"answerNote":11,"images":21,"explanation":23,"withheld":11,"deepChars":24,"freq":25,"indexable":26},"arch-106180-801-0102-009","architect","106180",106,"建築結構",9,null,"圖(a)所示懸臂梁在自由端承受 P 的集中載重，梁的斷面分別選用正方形斷面以及圓形斷面，如圖(b) 所示。當使用相同材料，且兩者的斷面積相等時，正方形斷面梁的最大變位為圓形斷面梁的多少倍？",{"A":14,"B":15,"C":16,"D":17},"1.047 P","0.955","0.846 A B E","1.182 L (b) (a)",[19],"B","single",[22],"\u002Fimages\u002Farch-106180-801-0102-009.png","本題考點：懸臂梁自由端受集中載重時的最大變位公式 δ = PL³\u002F(3EI)，以及「斷面積相等」條件下，正方形與圓形斷面慣性矩 I 的比較。\n【正解理由】依題圖(a)，懸臂梁 A 端固定、B 端自由，P 作用在自由端，最大變位發生在 B 點，δ = PL³\u002F(3EI)。兩根梁的 P、L 與材料彈性係數 E 都相同，所以最大變位只與慣性矩 I 成反比：δ方\u002Fδ圓 = I圓\u002FI方。設正方形邊長 a、圓形半徑 r，面積相等得 a² = πr²。正方形 I方 = a⁴\u002F12 = (πr²)²\u002F12 = π²r⁴\u002F12；圓形 I圓 = πr⁴\u002F4。相除 I圓\u002FI方 = (π\u002F4)\u002F(π²\u002F12) = 3\u002Fπ = 0.955。正方形梁的慣性矩較大、變位較小，是圓形梁的 0.955 倍，故選 B。\n【逐項排除】\n(A) 1.047：等於 π\u002F3 = 3.14159\u002F3 = 1.047，是「圓形梁變位 ÷ 正方形梁變位」，比值方向倒過來了；題目問的是正方形為圓形的多少倍，應取 I圓\u002FI方 而不是 I方\u002FI圓。\n(B) 0.955：正解。變位與 I 成反比，3\u002Fπ = 0.955，正方形變位較小，數值小於 1 也符合「同面積下正方形較勁」的直覺。\n(C) 0.846：等於 6\u002F(4√π) = 0.846，這是兩者「最大彎曲應力」的比值（下一題的答案），由斷面模數 S = I\u002Fc 決定；本題問變位，取決於慣性矩 I，兩個量不能混用。\n(D) 1.182：等於 4√π\u002F6 = 1.182，是最大應力比的倒數，量（應力而非變位）與比值方向兩處都不符。\n【演算步驟】\n① 公式：δmax = PL³\u002F(3EI)，P、L、E 相同 → δ方\u002Fδ圓 = I圓\u002FI方。\n② 面積相等：a² = πr²（長度²）。\n③ I方 = a⁴\u002F12 = π²r⁴\u002F12（長度⁴）；I圓 = πr⁴\u002F4（長度⁴）。\n④ δ方\u002Fδ圓 = (π\u002F4)\u002F(π²\u002F12) = 12\u002F(4π) = 3\u002Fπ = 3\u002F3.14159 = 0.955。\n【記憶點】變位看 I、應力看 S；同面積時正方形 I 是圓形的 π\u002F3 倍，所以正方形變位是圓形的 3\u002Fπ = 0.955 倍。",526,1,true,[28,31,34,37,40,43],{"number":29,"session":7,"stem":30},6,"臺灣位處強震帶，高層建築的結構系統配置原則，下列何者較為不佳？",{"number":32,"session":7,"stem":33},7,"高層建築結構常採用核心結構（core structure）系統，當核心尺度不足，可增…",{"number":35,"session":7,"stem":36},8,"下列何種系統為型抗結構（form-resistant structure）？",{"number":38,"session":7,"stem":39},10,"承上題，正方形斷面梁的最大應力為圓形斷面梁的多少倍？",{"number":41,"session":7,"stem":42},11,"依據「建築物耐震設計規範」之耐震設計目標，下列何者錯誤？",{"number":44,"session":7,"stem":45},12,"某十層樓建築物，其各層的質量皆相等，且各層的勁度與樓高亦相等。不考慮阻尼影響，在地震…",[47,50,53,56,59],{"number":10,"session":48,"stem":49},"114180","某等跨距構架活載重 w 全部負載、間隔負載如圖(a)、圖(b)所示，若活載重大於靜載…",{"number":10,"session":51,"stem":52},"113190","依「建築物耐震設計規範」，下列有關地震加速度反應譜之敘述何者正確？",{"number":10,"session":54,"stem":55},"112190","一個微小的矩型六面體受到純剪力作用，下列敘述何者正確？",{"number":10,"session":57,"stem":58},"111180","有一長度為 L，直徑為 d 之圓桿，彈性模數為 E，波松比（Poisson’s ra…",{"number":10,"session":60,"stem":61},"110180","下圖結構的穩定與可定性質為何？ L L L L",1791863211895]