矩陣力法
按力法的基本原理,以矩陣為數學工具,計算結構的內力和位移的方法,是結構矩陣分析方法中的一種。
正文
結構矩陣分析方法需將結構離散成有限數目的單元進行計算。矩陣力法中常用的單元形式為簡支式和懸臂式,這兩種單元較為簡單,其中尤以簡支式為常見。當單元承受非結點荷載時,可用等效結點荷載代替。其方法是將單元間的分界結點視為固定求出固端反力,然後反其向作用在結點上。
矩陣力法的基礎是力法
計算超靜定結構時要選取基本體系和基本未知力。選取的方法有兩種:一種是根據結構的具體情況由計算者選取,並在人為選定的基本體系的基礎上計算;另一種是把力法和線性代數中關於秩的知識結合起來,先建立結點平衡方程式,然後利用約當消去法,使多餘的基本未知力自動分離出來,這種分析方法稱為秩力法。由於前一方法與力法結合較為緊密,故較易了解和常用。
將原有荷載和基本未知力均視為外力時,可以得出結點作用力列矩陣、結構基本未知力列矩陣
與單元基本未知力(桿端力)列矩陣
的關係式如下:
=
P
+
X
(1)
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單元基本未知力
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=
m
(2)
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=
(3)
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根據上面三式,可以得到
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(9)
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(10)
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用矩陣力法求靜定結構的位移時,由公式(4)令基本未知力X=0,即可得靜定結構結點荷載方向位移的公式為
(11)