5-1 综合习题(推导方式)

UNIT 1 综合习题(推导方式)

习题要求：

(1) 要求完成UNIT 1.1基本习题(至少完成10个)，可以选择

完成UNIT 1.2 研讨习题

(2) 每位同学独立完成；

(3) 请手写书面完成，交手写稿，不要打印稿；

(4) 跟随课堂进度完成相应的习题，在课程结束时一并上交。

UNIT 1.1 基本习题

1.1(1) 分别推导出如题图1-1(1)(a)和题图1-1(1)(b)所示的一维锥形单元的单元刚度矩阵。（提示：利用位移插值的形状函数来描述题图1-1(1)(a)中宽度变化，题图1-1(1)(b)中直径为线性变化规律。）

k

(a) (b)

题图1-1(1)

1.1(2) 如题图1-1(2)所示，杆长为10cm ，采用推导的方式计算下列模型中的应力

(a)采用一个线性杆单元； (b)采用两个线性杆单元。

A=900mm 2

题图1-1(2) 题图1-1(3)

1.1(3) 如题图1-1(3)所示的等截面杆在其右半部分承受均布的轴向载荷，采用推导的方式计算杆上的应力分布情况，要求用罚函数法处理边界条件。

1.1(4) 如题图1-1(4)所示的受局部载荷作用的梁结构，左端为悬臂支撑，右端为弹簧支撑，采用推导的方式计算点1的位移量和点2的支反力。

E=200000MPa

题图1-1(4)

1.1(5) 如题图1-1(5)所示的梁结构，左端为悬臂支撑，点2的位置为铰连接，右端为铰支撑，采用推导的方式分析该结构，给出点2的位移量和点3的支反力。

E=200000MPa

题图1-1(5)

1.1(6)

如题图1-1(6)所示的平面梁，其AB 两端固定，由于安装误差使B 端垂直下移2mm ，采用推导的方式求解该问题，并给出支反力。

E=200000MPa I=150000mm 4

A

题图1-1(6)

1.1(7) 如题图1-1(7)所示一个两端固定的杆件承受变化的体力，假设位移场为

，采用最小势能原理求解相应的位移场和应力场2012u a a x a x =++()u x ()x σ。

题图1-1(7) 题图1-1(8)

1.1(8) 如题图1-1(8)所示对于一个平面悬臂梁，验证以下函数可以作为该问题的试函数，采用最小势能原理来求解该问题。

()02()1cos x

l v x a π=??

1.1(9) 在平面应变问题中，其应力状态为20MPa,

10MPa, 20MPa x y xy σστ==?=?，

材料的参数为 ；求垂直方向上应力5

2.110MPa,0.3E μ=×=z σ的值。

1.1(10) 有平面应力状态的几种受力状态如下，若所采用材料的单向拉伸的许用应力为

150MPa, 试用最大剪应力准则、最大畸变能准则计算各种情况的安全系数。

(1) 50MPa,50MPa,50MPa;x y xy σστ=== (2) 40MPa,80MPa,50MPa;x y xy σστ==?=? (3) 40MPa,80MPa,0;x y xy σστ=?== (4)

0,0,50MPa;x y xy σστ===?

1.1(11) 题图1-1(11)给出4种建模方式，讨论建模方法的正确性，若有错误，给出解决的方案。

题图1-1(11)

1.1(12) 如题图

1-1(12)所示为由两个三角形单元组成的平行四边形，

题图1-1(12)

已知单元1按局部编码的单元刚度矩阵是

,,i j m (1)

K

(1)

806

626160120413.5 4.57.5 4.513.5 1.5 1.5.9.5 1.55.5sym ???????????????=????

??????

????

K

试按局部坐标的编码写出单元2的单元刚度矩阵，并进行整体刚度矩阵的组装。

(2)

K

1.1(13) 已知一个弹性结构组装后的刚度矩阵为

3210012106.410MN/m 01210011?????????=××

??????

???

K 对应于该刚度矩阵，所求出的节点位移为；求该结构的应变能。

[0.00330.00580.00750.0085]m T

=?q

1.1(14) 如题图1-1(14)所示的结构，给出无量纲的参数，试用两个矩形单元求解载荷作用点处的位移。

E=1, t=1

题图1-1(14)

1.1(15) 对于如图1-1(15)所示的4个一维单元的应力分布，分别采用最小二乘方法、基于长度的加权平均方法以及直接平均方法进行节点的应力修匀，并比较计算结果。

题图1-1(15)

1-1(16) 如题图1-1(16)所示的结构处于一个温度升高的变化中，，外载为

材料及其参数也在图中给出，A 为横截面积，E 为弹性模量，α为线

膨胀系数。求位移场、应力和支座反力，用推导的方法求解该题。

80 T Δ=D C 2360kN,75kN,F F ==

题图1-1(16) 三杆结构与参数

UNIT 1.2 研讨习题(可选择)

1.2(1) 如题图1-2(1)所示结构，有两种边界条件，采用推导的方式计算不同边界条件下支点的支反力。

40mm

40mm

(a) (b)

题图1-2(1)

1.2(2) 图1-2(2)所示为平面梁结构，采用推导的方式计算梁的内力以及A 和C 点的支反力。

题图1-2(2)

1.2(3) 如题图1-2(3)所示，有一个电阻应变花的测量结果为

250 , 750 , 480A B C εμεμε===μ=

若材料的参数为，计算该点的主应力及其方向。 200 , 0.3a

E GP μ=

题图1-2(3)

1.2(4) 若一个平面变形体在受力后的位移场为

2232

3

(,)(648)100.6(,)(642)100.5u x y x y xy y cm v x y x y y cm

??=?+++×+=?+?×?

求出位于点 1 , 0x y ==处的应变分量,,x y xy εεγ，分析该位移场的刚体位移，并给出图示。

1.2(5) 如题图1-2(5)所示的6种应力状态，试用最大剪应力准则、最大畸变能准则、最大拉应力准则来判断和比较它们的危险程度。

题图1-2(5)

1.2(6) 如题图1-2(6)所示为一个单元的变形前与变形后的形貌，求出该单元的位移变形场

，由此计算(,),(,)u x y v x y ,,xx yy xy εεγ，并就应变分量的特征进行讨论。

题图1-2(6)

1.2(7) 如题图1-2(7)所示的结构分别采用了两种单元划分方式，采用推导的方式进行分析，讨论两种建模方式的结果有何不同，哪种方式的计算精度更高？为什么？

1

F=1

F=1

E=1,t=1

题图1-2(7)

1.2(8) 针对习题1.2(7)，若边界条件换成了题图1-2(8)所示的边界条件，进行同样的讨论。

F=1 F=1

E=1,t=1

题图1-2(8)

1.2(9) 对于如图1-2(9)所示的

6节点矩形单元，给出单元位移形状函数模式，并讨论是否满足收敛性条件。

y

题图1-2(9)

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