《材料力学》课程教学课件（PPT讲稿）第四章 弯曲内力 Internal forces in beams §4-4 剪力、弯矩与分布荷载集度间的关系（Relationships between load,shear force,and bending moment）

弯曲内力(Internalforces in beams)S4-4剪力、弯矩与分布荷载集度间的关系(Relationships betweenload,shearforce,andbending moment)一、弯矩、剪力与分布荷载集度间的微分关系(Differentialrelationshipsbetweenload,shearforce,andbendingmoment)设梁上作用有任意分布荷载V其集度q=q ()x规定g(x)向上为正将x轴的坐标原点取在梁的左端q(x)

1 (Internal forces in beams) 设梁上作用有任意分布荷载 其集度 §4-4 剪力、弯矩与分布荷载集度间的关系 (Relationships between load,shear force,and bending moment) 一、弯矩、剪力与分布荷载集度间的微分关系 (Differential relationships between load,shear force,and bending moment) q = q (x) 规定 q (x)向上为正. 将 x 轴的坐标原点取在梁的左端. x y q(x) F M

弯曲内力nternal forces in beamsXyM假想地用坐标为x和x+dxmx的两横截面m-m和n-n从梁中取出dx微段m-m截面上内力为Fs(x)，M(x)q(x)dxxx+dx截面处则分别为mnM(x)+dM(x)Fs(x)Fs(x) + dFs(x) , M(x) +作dM(x）.由于dx很小，略去M(x)q(x) 沿 dx 的变化Fs(x)+dFs(x)11mq(x)

2 (Internal forces in beams) x y q(x) F M FS (x) M(x) FS (x)+dFS (x) M(x)+dM(x) 假想地用坐标为x 和 x + dx 的两横截面m-m 和 n-n从梁中取 出 dx 微段. m m n n q(x) C x + dx 截面处 则分别为 FS (x) + dFS (x) ， M(x) + dM(x) . 由于dx 很小，略去 q(x) 沿 dx 的变化. m-m 截面上内力为 FS (x) , M(x) n x m m n dx

弯曲内力nternal forces in beams写出微段梁的平衡方程EF, =0Fs(x)-[Fs(x) + dFs(x)]+ q(x)dx = 0dFs(x)得到q(x)ZMc=0dxdx[M(x)+ dM(x) ]- M(x) - Fs(x) dx - g(x)dx2mnM(x)+dM(x)略去二阶无穷小量即得Fs(x)格M(x)fti-.-f-dM(x)= Fs(x)Fs(x)+dFs(x)dxnmq(x)

3 (Internal forces in beams) 0 2 d [ ( ) d ( ) ] ( ) ( ) d ( )d 0 + − − S − =  = x M x M x M x F x x q x x MC FS (x) M(x) FS (x)+dFS (x) M(x)+dM(x) m m n n q(x) C 写出微段梁的平衡方程 Fx = 0 FS (x) − [FS (x) + dFS (x)]+ q(x)dx = 0 得到 ( ) d d ( ) S q x x F x = 略去二阶无穷小量即得 ( ) d d ( ) FS x x M x =

弯曲内力nternal forces in beamsdFs(x)q(x)dxdM(x)Fs(x)dxd'M(x)g(x)dx?公式的几何意义(1)剪力图上某点处的切线斜率等于该点处荷载集度的大小(2)弯矩图上某点处的切线斜率等于该点处剪力的大小：(3）根据g(x)>0 或g(x)< 0 来判断弯矩图的凹凸性

4 (Internal forces in beams) 公式的几何意义 （1）剪力图上某点处的切线斜率等于该点处荷载集度的大小; （2）弯矩图上某点处的切线斜率等于该点处剪力的大小; ( ) d d ( ) S q x x F x = ( ) d d ( ) S F x x M x = ( ) d d ( ) 2 2 q x x M x = （3）根据 q(x) ＞ 0 或 q(x) ＜ 0 来判断弯矩图的凹凸性

弯曲内力(nternal forces in beams二、q(x）、Fs(x)图、M（x）图三者间的关系(Relationships between load, shear force, and bendingmoment diagramsdFs(x)1.梁上有向下的均布荷载，即g(x)<0q(x)dxFs(x）图为一向右下方倾斜的直线dM(x)= Fs(x)M(x）图为一向上凸的二次抛物线dxM(x)Fs(x)d°M(x)q(x)dr?00xX

5 (Internal forces in beams) M(x) 图为一向上凸的二次抛物线. FS (x) 图为一向右下方倾斜的直线. x FS (x) O 二、q(x)、FS(x)图、M（x）图三者间的关系 (Relationships between load, shear force, and bending moment diagrams) 1.梁上有向下的均布荷载,即 q(x) < 0 M(x) ( ) d d ( ) S q x x F x = ( ) d d ( ) S F x x M x = ( ) d d ( ) 2 2 q x x M x =