STATICALLYINDETERMINATE STRUCTURES 8 12-2 SOLVE STATICALLY INDETERMINATE STRUCTUREBY THE FORCE METHOD I Thoughts of the force method (explain by examples P Example 1 El is a constant in the beam shown in the figure. Try to determine its constraint reactions, plot the (a) B bending-moment diagram and determine the deflection of the middle point C on the 2 beam Solution: (1 Determine the number of redundant constraint reactions. (one) (b) (2 Select the redundant constraint and take B A C X it out, substitute it by its reaction. Write out the compatibility equation of deformation. See Fig. (b)
11 §12–2 SOLVE STATICALLY INDETERMINATE STRUCTURE BY THE FORCE METHOD 1、Thoughts of the force method（explain by examples） Solution：①Determine the number of redundant constraint reactions.(one) ②Select the redundant constraint and take it out, substitute it by its reaction. Write out the compatibility equation of deformation. See Fig.(b) C 2 l P A B 2 l (a) P A B C X1 (b) Example 1 EI is a constant in the beam shown in the figure.Try to determine its constraint reactions, plot the bending-moment diagram and determine the deflection of the middle point C on the beam
§122用力法解静不定结构 力法的基本思路(举例说明) P [例1]如图所示,梁E为常数。 (a) B 试求支座反力,作弯矩图,并 求梁中点的挠度。 2 解:①判定多余约束反力的数目 (b) ②选取并去除多余约束,代 B A C X 以多余约束反力,列出变形 协调方程,见图(b)。 12
12 §12–2 用力法解静不定结构 一、力法的基本思路（举例说明） 解：①判定多余约束反力的数目 （一个） ②选取并去除多余约束，代 以多余约束反力，列出变形 协调方程，见图(b)。 C 2 l [例1 ] 如图所示，梁EI为常数。 试求支座反力，作弯矩图，并 求梁中点的挠度。 P A B 2 l (a) P A B C X1 (b)
STATICALLYINDETERMINATE STRUCTURES △B=△1x,+△1P=0 Compatibility equation of deformation P ③ CalculateAin and△x. by the energy method (c) C B A From Mohr's theorem we get: (Fig.cv d、e) B △PEI2 -P(x-x-dx 2 Vx Xi 5P13 (e) B 48EI X xxx= El 3EⅠ 13
13 1 1 0 1 B = X + P = Compatibility equation of deformation ③Calculate and by the energy method 1P 1X1 P A B C (c) x (d) x A B X1 A B 1 x (e) From Mohr’s theorem we get: (Fig.c、 d、e) EI Pl x x l P x EI l P l 48 5 ) d 2 ( 1 3 2 1 =− = − − EI X l X x x x EI l X 3 d 1 3 1 0 1 1 = 1 =
△。=△y+△,=0变形协调方程 P ③用能量法计算△p和△1x1 C B A 由莫尔定理可得(图c、d、e) △ P(-)xdx B E 2 Vx Xi 5P1 48E (e) B △ X xxx= El 3EⅠ
14 1 1 0 1 B = X + P = 变形协调方程 ③用能量法计算 1P 和 1 1X P A B C (c) x (d) x A B X1 A B 1 x (e) 由莫尔定理可得(图c、d、e) EI Pl x x l P x EI l P l 48 5 ) d 2 ( 1 3 2 1 =− = − − EI X l X x x x EI l X 3 d 1 3 1 0 1 1 = 1 =
STATICALLYINDETERMINATE STRUCTURES A Determine redundantreactions Substituting the above result into the 11P compatibility equation of deformation we P get X 1 5P13 0X1 (f)( 3 B C BE 48EI 16 5网 Determine other constraint reactions 16 16 The reactions at the end a may be found 5Pl out by the static equilibrium equations Their magnitudes and directions are shown in Fig. (g) pLot the bending-moment diagram as 3Pl shown in Fig. g) 16 ( Determine the deflection at the middle point of the beam 15
15 ④Determine redundant reactions Substituting the above result into the compatibility equation of deformation we get 0 48 5 3 3 3 1 − = EI Pl EI X l X P 16 5 1 = ⑤Determine other constraint reactions. C P A ( f ) B 16 5P 16 11P 16 3Pl ⑥Plot the bending-moment diagram as shown in Fig.(g). (g) + – 16 3Pl 32 5Pl ⑦Determine the deflection at the middle point of the beam. The reactions at the end A may be found out by the static equilibrium equations. Their magnitudes and directions are shown in Fig.(f)