Mechanics of Materials - APS
1. Basic
Obeject : Deformed solid
internal force : interaction created by inner part change
- Axial force : $F_N$
- Shear force : $F_{sy},F_{sz}$
- torque : $T$
- bending moment : $M_y,M_z$
Stress : section, Pa
- normal stress : $\sigma$
- shear : $\tau$
- full
Strain
normal strain : $\varepsilon$, no unit
shear strain : $\gamma$
Hooke theory : $\sigma=E\varepsilon$
- E : Elastic Modulus
Shear hooke theory : $\tau=G\gamma$
- G : shear elastic modulus
Component
- rod piece
- block
- shell
- board
basic deformation
- stretch & compress
- shear
- torsion
- bend
2 Axial tension & compression
2.1 Concept
along the axis
make axis length greater or less
$F_N$
positive : out
negative : in
graph
2.2 Strain
$$\sigma=\dfrac{F_N}{A}$$
- Pa
slope section
Pull Rod Strength Condition
- ultimate stress : break down the work piece
- allowable stress : $[\sigma]=\dfrac{\sigma}{n}$
- n : safety factor
- strength condition : $\sigma_{max}\le[\sigma]$
utilize
- strength check
- section size design
- make sure external load
2.3 Deformation
axial : $\Delta L=L’-L$
- linear strain : $\varepsilon=\dfrac{\Delta L}{L}$
$$\Delta L=\dfrac{F_NL}{EA}$$
- EA : extensional rigidity
- $\Delta l$ : stretch +, compress -
lateral : $\Delta b=b_1-b$
- lateral linear strain : $\varepsilon’=\dfrac{\Delta b}{b}$
- $\varepsilon’=-\mu\varepsilon$
- $\mu$ : poisson ratio
2.4 Mechanical Properity
elastic range OA
- $\sigma_e$ : elastic limit
yield range BC
- $\sigma_s$ : yield limit
hardening range CD
- $\sigma_B$ : ultimate strength
necking range DE
cold hardening
- elastic strain
- plastic strain
over yield range, unload and load, make $\sigma_e$ higher
2.4.1 plasticity
$\delta=\dfrac{\Delta l}{l_0}\times 100\%$
- > 5,% plastic material
- <5%, brittleness material
2.4.2 Tensile test
Low-carbon steel
cast iron
2.5 Statically Indeterminate
unknown force more than statics equation, caused by Redundant Restraint
3 Shear
external force place at two side of material, same modulus, opposite direction, close active line
Strength Condition
utilize
- strength check
- section size design
- make sure external load
shear
$$\tau=\dfrac{F_S}{A}\le[\tau]$$
- $[\tau]$ : allowable shear stress
extrusion
$$\sigma_{bs}=\dfrac{F_{bs}}{A_{bs}}\le[\sigma]_{bs}$$
4 Torsion
- +, out
- -,in
- graph
4.1 Shear stress
$\varepsilon=0,\sigma=0$
thin tube : $T=\int \tau dAr_0=\int\limits_0^{2\pi}\tau r^2 td\alpha=\tau r^2_0t2\pi$
$$\tau=\dfrac{T}{2\pi r^2_0t}$$
4.2 Shear Hooke Principle
$$\tau=G\gamma=G\rho\dfrac{d\varphi}{dx}$$
with $T=\int_AdA\tau_\rho=G\dfrac{d\varphi}{dx}\int_A\rho^2dA$
let $I_A=\int_A\rho^2dA$
so $\dfrac{d\varphi}{dx}=\dfrac{T}{GI_p}$
then $\tau_\rho=\dfrac{T\rho}{I_\rho}$
$$\tau_{max}=\dfrac{T}{I_\rho}\rho_{max}=\dfrac{T}{W_\rho}$$
- $I_p$ : polar moment of inertia , m4
- $W_p$, anti torsion section modulus, m3
| $I_p$ | $W_p$ | |
|---|---|---|
| solid circle | $\dfrac{\pi d^4}{32}$ | $\dfrac{\pi d^3}{16}$ |
| hollow circle | $\dfrac{\pi D^4}{32}(1-\alpha^4)$ | $\dfrac{\pi D^3}{16}(1-\alpha^4)$ |
4.3 Shear stress equal
the vertical plane have a pair of shear stress with same modulus
4.4 Strength Check
$\tau_{max}=\dfrac{T_{max}}{W_p}\le[\tau]$
4.5 Deformation
$\dfrac{d\varphi}{dx}=\dfrac{T}{GI_p}=\theta$
- $\theta$ : angle of torsion
so $\varphi=\int_L\dfrac{T}{GI_p}dx$
Stiffness check
$\theta_{max}=\dfrac{T_{max}}{GI_p}\le[\theta]$
4.6 Material break
- low-carbon steel : cross section
- cast iron : 45 degree spiral
4.7 Rectangular section
- vertex = 0
- side : parallel
long rectangular section
5 Bending stress
- force vertical to beam, at the mid
5.1 Type
- cantilever
- simply supported beam
- overhanging beam
5.2 Shear force and bending moment
5.2.1 Equation
$$\dfrac{dF_s(x)}{dx}=q(x)\\dfrac{dM(x)}{dx}=F_s(x)\\dfrac{d^2M(x)}{dx^2}=q(x)$$
5.2.2 Graph
5.2.3 Superposition
6 Beam stress
6.1 Normal stress
6.1.1 Pure bending
suppose all the landscape and portrait plane still plane after deformation
bending is actually each section rotate around the neural surface, one side compress and the other strech
strain
$$\varepsilon=\dfrac{y}{\rho}$$
so $\sigma=E\varepsilon=\dfrac{Ey}{\rho}$
6.1.2 Statics balance condition
$$\sigma=\dfrac{My}{I_z}$$
positive, up
negative, down
$\sigma_{max}=\dfrac{My_{max}}{I_z}=\dfrac{M}{W_z}$
$W_z$ : section modulus in bending
| | $I_z$ | $W_z$ |
| ————- | ——————————— | ——————————— |
| solid circle | $\dfrac{\pi d^4}{64}$ | $\dfrac{\pi d^3}{32}$ |
| hollow circle | $\dfrac{\pi D^4}{64}(1-\alpha^4)$ | $\dfrac{\pi D^3}{32}(1-\alpha^4)$ |
6.1.3 Strength Condition
$\sigma_{max}=\dfrac{M_{max}}{W_z}\le[\sigma]$
6.2 Shear stress
$$\tau=\dfrac{F_ZS_z^*}{I_Zb}$$
- $S_Z^*=y_c^*A^*$
- max mid : I-beam, round beam
Strength Condition
normal in the neural surface
$$\tau_{max}=\dfrac{F_{max}S_{zmax}^*}{I_zb}\le[\tau]$$
7 Bending deformation
- Deflection curve : deformation axis curve
- Deflection $y$ : section centroid vertical movement
- angle $\theta$ : the section rotate angle round neural axis
- angle equation $\theta(x)=\dfrac{dy}{dx}=y’$
7.1 Deflection curve differential function
$\dfrac{1}{\rho(x)}=\dfrac{M(x)}{EI_z}\\dfrac{1}{\rho}=\pm y’’$, so $EIy’’=\pm M$, usually
$$EIy’’=-M(x)$$
- Boundary condition
- fixed end : $y=0,\theta=0$
- hinged shoe : $y=0$
- Continuity condition
- $y_1=y_2, \theta_1=\theta_2$
Superposition
7.2 Strength Condition
$$y_{max}\le[\delta]\\theta_{max}\le[\theta]$$
- strength check
- section design
- external load
7.3 Indeterminate
make the constraint to a force, and let the y here to be 0
7.4 Beam Combination
cut at joint, relationship condition
8 Stress State
8.1 Stress element
- main plane : $\tau=0$
- main stress : at main plane
- $\sigma_1\ge\sigma_2\ge\sigma_3$
8.2 State type
- 3-d
- 2-d
- single stress
- pure shear
8.3 Analysis
8.3.1 Analytical
$\sigma$ max when $\dfrac{d\sigma_\alpha}{d\alpha}\Big\lvert_{\alpha_0}=0$
$$\sigma_{max/min}=\dfrac{\sigma_x+\sigma_y}{2}\pm\sqrt{(\dfrac{\sigma_x-\sigma_y}{2})^2+\tau^2_x}$$
$\tau$ max , when $\dfrac{d\tau_\alpha}{d\alpha}\Big\lvert_{\alpha_1}=0$
$$\tau_{\max/\min}=\pm\sqrt{(\dfrac{\sigma_x-\sigma_y}{2})^2+\tau^2_x}$$
$\alpha_1=\alpha_0+45^\circ$
8.3.2 Graphical
- stress circle
3d stress state
8.3 General Hooke Principle
8.4 Strength Theory
Maximal pulling stress (1st)
$$\sigma_1\le\dfrac{\sigma_b}{n}=[\sigma]$$
Maximal pulling strain (2nd)
$$\varepsilon=\dfrac{1}{E}[\sigma_1-\mu(\sigma_2+\sigma_3)]\le[\sigma]$$
Maximal shear stress(3rd)
$$\tau_{max}=\dfrac{1}{2}(\sigma_1-\sigma_3)\le[\sigma]$$
Shape deformation ratio energy
$v_d=\dfrac{1+v}{6E}[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2]$
$$\sqrt{\dfrac{1}{2}[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2]}\le[\sigma]$$
9 Deformation Combination
use Superposition
9.1 Oblique bending
- strength condition
deformation calculation
9.2 Pulling & Bending
strength condition
$$\sigma_{max}^{top}=\dfrac{M_{zmax}}{W_z}+\dfrac{F_N}{A}\le[\sigma]\\sigma_{max}^{bottom}=\dfrac{M_{zmax}}{W_z}+\dfrac{F_N}{A}\le[\sigma]$$
Eccentrical pull
9.3 Torsion & Bending & Pull
10 Energy Method
10.1 External Force Work
$$W=\dfrac{1}{2}\sum F_i\delta_i$$
10.2 Strain energy
- pull
shear
torsion
bending
combination
$$U=\int\dfrac{F^2_N(x)dx}{2EA}+\int\dfrac{M^2_t(x)dx}{2GI_p}+\int\dfrac{M^2(x)dx}{2EI}$$
10.3 complementary work
Displacement Calculation
10.4 Reciprocal principle
10.5 Castigliano’s theorem
CT1
$$\dfrac{\partial U}{\partial \delta_i}=F_i$$
Complementary theory
$$\delta_i=\dfrac{\partial U^*}{\partial F_i}$$
CT2
$$\delta_i=\dfrac{\partial U}{\partial F_i}$$
CT2 for indeterminate problem
11 Press Stability
- example
- flexibility $\lambda$
- radius of inertia i
11.1 critical force
11.2 Calculation
safety factor
the p-multiplier method
12 Dynamic load
static acceleration : $\sigma_d=\dfrac{F_{Nd}}{A}=(1+\dfrac{a}{g})\gamma x$
static angular velocity
- $F=ma_n=\dfrac{\gamma AL}{g}w^2\dfrac{D}{2}$
- $q_d=\dfrac{ma_n}{L}$
- $\sigma_d=\dfrac{F_nd}{A}=\dfrac{q_dD}{2A}=\dfrac{\gamma v^2}{g}$
shock : energy conservation
free fall
- $\Delta_d=K_d\Delta_{st}=(1+\sqrt{1+\dfrac{2h}{\Delta_{st}}})\Delta_{st}$
- $\sigma_d=K_d\dfrac{Q}{A}$
horizontal shock
$K_d=\sqrt{\dfrac{v^2}{g\Delta_{st}}}$