轴向拉伸与压缩时的变形11.ppt

第四节第四节 轴向拉伸与压缩时的变轴向拉伸与压缩时的变形形 胡克定律胡克定律一、纵向变形一、纵向变形 〔一〕绝对变形 杆件长度的伸长〔缩短〕量称为绝对变形〔二〕相对变形 单位长度的变形称为相对变形，也称为线应变沿轴线方向单位长度的变形称为纵向相对变形或纵向线变形da)F Fdb)〔4–1〕〔4–2〕F Fdc)图4-8 二、横向变形二、横向变形〔一〕绝对变形〔一〕绝对变形 杆件横向尺寸的缩小〔增大〕量称为横向绝对变杆件横向尺寸的缩小〔增大〕量称为横向绝对变形，假设以形，假设以 △△d 表示，那么表示，那么 △△d =d1 –d 〔二〕相对变形〔二〕相对变形 横向单位长度的变形称为横向相对变形或横向线应横向单位长度的变形称为横向相对变形或横向线应应变应变，若以 表示，则〔4–3〕三、泊松比三、泊松比 当应力不超过某一限度时，同一种材料的横向线应变与纵向线应变之比的绝对值为一常数，即式中， 为横向变形系数横向变形系数或泊松比，是一个量纲为1的量通常由试验测得,工程上常用材料的泊松比列于下表。

常用材料的常用材料的 E E ，， 值值 材 料 名 称 E／GPa 低碳钢 196～216 0.25～0.33 合金钢 186～216 0.24～0.33 灰铸铁 115～157 0.23～0.27 铜合金 72.6～128 0.31～0.42 铝合金 70 0.33〔4–4〕 四、胡克定律四、胡克定律 只要应力不超过某一极限值时，杆的轴向变形与轴向只要应力不超过某一极限值时，杆的轴向变形与轴向载荷成正比、与杆的长度成反比、与杆的横截面面积成反载荷成正比、与杆的长度成反比、与杆的横截面面积成反比。

比这一关系称为胡克定律，即胡克定律，即∝引进比例常数E ，那么有由于轴向拉压时有F = FN ,故上式可改为 〔4–5〕式中，E 为材料的弹性模量弹性模量，单位常用GPa 对同一材料，E 为常数由式(4–5)可知，受力和长度相同的杆件，绝对变形 和 的乘积成反比，该乘积愈大，变形就愈小它反映了杆件抵抗拉伸(压缩)变形的能力，故 称为杆的抗拉抗拉( (压压) )刚度将 、 代入式(4–5)，便得(4–6)式(4-6)是胡克定律的又一表达形式：当应力不超过某一当应力不超过某一极限时，应力与应变成正比极限时，应力与应变成正比  例例 一钢制阶梯杆受力如图一钢制阶梯杆受力如图4-9a所示，其横截面面积所示，其横截面面积分别为分别为ACD = 300mm2 ，，AAB =ABC =500mm2 , 材料的弹材料的弹性模量性模量E =200GPa 试求：(1)杆的总变形杆的总变形 (2)杆的最大纵向线应变杆的最大纵向线应变图4-9ABCD112210010010030kN10kNa)(+)(-)F FN20kN10kNxb) 解解 (1)画轴力图用截面法求得截面1-1和2-2上的轴力分别为画出杆的轴力图〔图4-9b) (2)计算各杆的变形将阶梯杆分为AB、BC、和CD 段，应用胡克定律分别求出各段杆的变形量为AB 段BC 段CD 段 (3)计算杆的总变形 杆的总变形等于各段杆的变形量之和整个杆件缩短了0.0067mm(4) 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