[{"data":1,"prerenderedAt":62},["ShallowReactive",2],{"q-106180-arch_structure-26":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-106180-801-0102-026","architect","106180",106,"建築結構",26,null,"下圖簡支梁之跨度為 L。已知材料之彈性係數為 E，梁斷面之慣性矩為 I。今於跨度中央處施加一集中荷重 P，則該梁於跨度中央處之撓度應為：",{"A":14,"B":15,"C":16,"D":17},"PI\u002F(48EL3) P","PL\u002F(48EI3)","PE\u002F(48IL3) E,I","PL3\u002F(48EI) L\u002F2 L\u002F2",[19],"D","single",[22],"\u002Fimages\u002Farch-106180-801-0102-026.png","本題考點：簡支梁跨度中央受集中載重時的中點撓度公式 δ = PL³\u002F(48EI)，可用彎矩面積法推導，並以因次檢核排除其他選項。\n【正解理由】依題圖，簡支梁左端鉸支、右端滾支，P 作用在跨度中央，兩端反力各 P\u002F2，中點最大彎矩 Mmax = (P\u002F2)(L\u002F2) = PL\u002F4。由對稱，中點切線水平；以彎矩面積法，支承 A 的轉角等於半跨 M\u002FEI 圖面積 θA = (1\u002F2)(L\u002F2)(PL\u002F4)\u002FEI，中點撓度等於 A 點切線在中點的偏離量扣掉半跨 M\u002FEI 面積對中點的一次矩：δ = θA × (L\u002F2) − θA × (L\u002F6) = θA × (L\u002F2 − L\u002F6)，整理得 δ = PL³\u002F(48EI)，故選 D。\n【逐項排除】\n(A) PI\u002F(48EL³)：撓度的單位必須是長度。PI\u002F(EL³) 的因次為 力 × 長度⁴ ÷（力\u002F長度² × 長度³）= 長度³，量綱不符；I 與 L³ 的位置寫反了，正確應為 L³ 在分子、I 在分母。\n(B) PL\u002F(48EI³)：因次為 力 × 長度 ÷（力\u002F長度² × 長度¹²），不是長度；跨度的三次方與慣性矩的位置錯置。\n(C) PE\u002F(48IL³)：E 應在分母而放在分子，材料越硬撓度反而越大，與物理常識相反；因次也不是長度。正確是剛度 EI 越大撓度越小。\n(D) PL³\u002F(48EI)：正解。因次為 力 × 長度³ ÷（力\u002F長度² × 長度⁴）= 長度。\n【演算步驟】\n① 反力：RA = RB = P\u002F2（力）；Mmax = (P\u002F2) × (L\u002F2) = PL\u002F4（力 × 長度）。\n② θA = (1\u002F2) × (L\u002F2) × (PL\u002F4)\u002FEI（弧度）。\n③ δ中 = θA × (L\u002F2 − L\u002F6)（長度）。\n④ 係數：(1\u002F2) × (1\u002F2) × (1\u002F4) × (1\u002F2 − 1\u002F6) = 1\u002F48 → δ中 = PL³\u002F(48EI)。\n【記憶點】簡支梁中點集中載重：Mmax = PL\u002F4、δmax = PL³\u002F(48EI)；撓度一定是「載重 × 跨度三次方 ÷ EI」。",409,1,true,[28,31,34,37,40,43],{"number":29,"session":7,"stem":30},23,"如圖所示桁架，桿件 ED 的內力為何？",{"number":32,"session":7,"stem":33},24,"下圖桿件之自由端 a 點受 2P 力作用，a 點垂直向下伸長量為 δ av = ，則…",{"number":35,"session":7,"stem":36},25,"有一矩形梁斷面，梁寬 30 cm，梁高 60 cm。已知材料之彈性係數為 20,00…",{"number":38,"session":7,"stem":39},27,"下圖之懸臂梁，端點 B 承受一集中載重 P，則 B 點之垂直位移為： PL2",{"number":41,"session":7,"stem":42},28,"桁架結構受力如下圖，桿件 EC 之軸向力為多少 kN？",{"number":44,"session":7,"stem":45},29,"剛架結構受力如下圖，接點 C 之內力矩為多少 kN-m？",[47,50,53,56,59],{"number":10,"session":48,"stem":49},"114180","下列何者載重分布，使懸臂梁固定端彎矩最大？ 2w 2w w 0 0 L L L (a…",{"number":10,"session":51,"stem":52},"113190","如圖所示之構架承受側力 P，兩柱分別高 L 及 2 L，柱頂皆為鉸接（pin），兩柱…",{"number":10,"session":54,"stem":55},"112190","下圖中之圖(i)所示為一單跨抗彎矩構架（moment-resisting frame…",{"number":10,"session":57,"stem":58},"111180","試分析下圖桁架結構，下列選項何者正確（T 為張力，C 為壓力）？ 10 m 10 m…",{"number":10,"session":60,"stem":61},"110180","一簡支梁跨距為 5 m，分別受到以下兩種載重形式：（A）跨距中央受到集中載重 10 …",1791863212017]