MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C3EAAD.52722540" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C3EAAD.52722540 Content-Location: file:///C:/ECB24D8E/Prikladna_mexanika-Shapin.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1251" ПРИКЛАДНАЯ МЕХАНИКА

Прикладная механика

"Несчастны те люди,
которым все ясно"
(Луи Пастер)

Уважаемые коллеги!

Вашему вниманию предлагается цикл лекций по учебной дисциплине "Прикладная механика". Он предназначен для знакомства и познания вопросов надежности элементов инженерных конструкций, изучение которых предусмотрено в цикле общеобразовательных дисциплин для студентов всех энергетических специальностей.

Цикл лекций построен со вставками самоопрос= ных тестов по теоретической механике, физике и высшей математике, без знания которых изучение предлагаемой дисциплины невозм= ожно. Собственная оценка Вашего остаточного ресурса знаний по этим предметам позв= олит Вам принять соответствующее решение по обращению к системе тестовых тренаже= ров. Особенно это касается так называемой эпюрной техники внутренних усилий, кот= орая является своеобразной базовой площадкой перед проведением расчетов конструк= ций по параметрам надежности.

Введение цветных обозначений в классификаци= и сил и напряжений включает дополнительно Вашу зрительную память и, как показывает опыт автора, в немалой степени способствует добротному изучению предмета.

Каждая лекция представляет собой законченное решение той или иной проблемы ПРИКЛАДНОЙ МЕХАНИКИ. В каждой лекции использу= ется собственная нумерация рисунков. В этих же целях максимального приближения электронного варианта к "натуральному" количество обозначений фор= мул сведено к минимуму при автономно-однозначном расположении в каждой лекции.<= /p>

С уважением= :

автор
ШАПИН Вадим Иванович,
технический оформитель
ЧЕРНОВА Галина Николаевна

Введение и основные понятия

Ключевые слова: Прочность. Жесткость. Устойчивость. Надежность. Деформирование. Ресурс. Отказ.

Постановка задачи. Прикладная механика - это наука, интегрирующая, с одной стороны циклы общеобразователь= ных дисциплин таких как: физика, математика, теоретическая механика, материаловедение, инженерная графика, а с другой стороны - это первая инженерная дисциплина, которая преподается студентам технических специальностей. Прикладная механика, в принципе, охватывает две дисциплины: сопротивление материалов и основы конструирования. Ниже излагается цикл лек= ций по прикладной механике с расстановкой акцентов на наиболее сложно воспринимаемой части курса - сопротивлению материалов.

Сопротивление материалов - наука о прочности, жесткости и надежности элементов инженерных конструкций. Методами сопротивления материалов ведутся практические расчеты и определяются необходимые, как говорят, надежные размеры деталей машин, различных констру= кций и сооружений.

Основные понятия сопротивления материалов опираются на законы и теоремы общей механики и в первую очередь = на законы статики, без знания которых изучение данного предмета становится практически невозможным.

В отличие от теоретической механики сопротивление материалов рассматривает задачи, где наиболее существенными являются свойства деформируемых тел, а законы движения тела, как жесткого целого, не только отступают на второй план, но в ряде случаев являются попр= осту несущественными.

Сопротивление материалов имеет целью создать практически приемлемые простые приемы расчета типичных, наиболее ча= сто встречающихся элементов конструкций. Необходимость довести решение каждой практической задачи до некоторого числового результата заставляет в ряде случаев прибегать к упрощающим гипотезам - предположениям, которые оправдываются в дальнейшем путем сопоставления расчетных данных с экспериментом.

Необходимо отметить, что первые заме= тки о прочности упоминаются в записках известного художника ЛЕОНАРДО Де ВИНЧИ, а начало науки о сопротивлении материалов связывают с именем знаменитого физика, математика и астронома ГАЛИЛЕО ГАЛИЛ= ЕЯ. В 1660 году Р.ГУК сформулировал закон, устанавливающий связь между нагрузко= й и деформацией: "Какова сила - таково и действие". В XVIII ве= ке необходимо отметить работы Л.ЭЙЛЕРА по устойчивости конструкций. XIX - XX в= ека являются временем наиболее интенсивного развития науки в связи с общим бурн= ым ростом строительства и промышленного производства при безусловно огромном вкладе ученых-механиков России.

Итак, мы будем заниматься твердыми деформированными телами с изучением их физических свойств.

Введем основные понятия, принимаемые при изучении дисциплины.

Прочность - это способность конструкции выдерживать заданную нагрузку, не разрушаясь.

Жесткость - способность конструкции к деформированию в соответствие с заданным нормативным регламен= том.

Деформирование - свойство конструкции изменять свои геометрические размеры и форму под действием внеш= них сил

Устойчивость - свойство конструкции сохранять при действии внешних сил заданную форму равновесия.

Надежность - свойство конструкции выполнять заданные функции, сохраняя свои эксплуатационные показатели в определенных нормативных пределах в течение требуемого промежу= тка времени.

Ресурс - допустимый срок службы изделия. Указывается в виде общего времени наработки или числа циклов нагружения конструкции.

Отказ - нарушение работоспособности конструкции.

Опираясь на вышесказанное, можно дать определение прочностной надежности.

Прочностной надежно= стью называется отсутствие отказов, связанных с разрушением или недопустимыми деформациями элементов конструкции.

На рис.1 приведена структура модели прочностной надежности. Она включает известные модели или ограничения, которые априорно накладываются на свойства материал= ов, геометрию, формы изделия, способы нагружения, а также модель разрушения. Инженерные модели сплошной среды рассматривают материал как сплошное и однородное тело, наделенное свойством однородности структуры. Модель матери= ала наделяется свойствами упругости, пластичности и ползучести.

Упругостью называется свой= ство тела восстанавливать свою форму после снятия внешних нагрузок.

Пластичностью называется свойство тела сохранять после прекращения действия нагрузки, или частично полученную при нагружении, деформацию.

Ползучестью называется свойство тела увеличивать деформацию при постоянных внешних нагрузках.=

Основными моделями формы в моделях прочностной надежности, как известно, являются: стержни, пластины, оболочки= и пространственные тела (массивы) (рис.2). Модели

нагружения содержат схема= тизацию внешних нагрузок по величине, характеру распределения (сосредоточенная или распределенная сила или момент), а также воздействию внешних полей и сред. =

После обоснованного выбора моделей формы, материала, нагружения переходят к непосредственной оценке надежности= с помощью моделей разрушения. Модели разрушения представляют собой уравнения, связывающие параметры работоспособности элемента конструкции в момент разрушения с параметрами, обеспечивающими прочность. Эти уравнения (условия) называют условиями прочности. Обычно рассматриваются в зависимости от услов= ий нагружения четыре модели разрушения:

  • стат= ические,
  • длит= ельно статические,
  • мало= цикловые,
  • уста= лостные.

Как уже отмечалось, изучение дисципл= ины невозможно без знания основ теоретической механики. Поэтому свой остаточный ресурс знаний рекомендую проверить по разделу "Статика", используя систему входных тестов.

Поскольку изучение сопротивления материалов базируется прежде всего на таких изве= стных понятиях как сила, пара сил, связи, реакции в связях, равнодействующая сист= ема внешних сил, то…

Вам рекомендуется решить простые задачи, указанные в ПРИЛОЖЕНИИ под разделом Т-1.=

Метод сечений для определения внутренних усилий

Ключевые слова: Внешние си= лы. Внутренние усилия (силовые факторы). Следящая система координат. Нормальная сила. Внутренние крутящие и изгибающие моменты. Поперечная сила.

Деформации рассматриваемого тела (элементов конструкции) возникают от прохождения внешней силы. При этом изменяются расстояния между частицами тела, что в свою очередь приводит к изменению сил взаимного притяжения между ними. Отсюда, как следствие, возни= кают внутренние усилия. При этом внутренние усилия определяются универсальным методом сечений (или метод Разреза).

Известно, что различают силы внешние= и силы внутренние. Внешние усилия (нагрузки) - это количественная мера взаимодействия двух различных тел. К ним относятся и реакции в связях. Внутренние усилия - это количественная мера взаимодействия двух частей одно= го тела, расположенных по разные стороны сечения и вызванные действием внешних усилий. Внутренние усилия возникают непосредственно в деформируемом теле.

На рис.1 приведена расчетная схема бруса с произвольной комбинацией внешней нагрузки образующую равновесную систему сил:

(1)

При этом, реакции связей определяются из известных уравнений равновесия статики тверд= ого тела:

,

(2)

,

,

где х0, у0, z0 - базовая система координат осей.

Мысленное разрезание бруса на две ча= сти произвольным сечением А (рис.1 a), приводит к условиям равновесия каждой из двух отсеченных частей (рис.1 б,в). Здесь {S'} и {S"}- внутренние усилия, возникающих соответственно в левой и правой отсеченных частях вследствие действия внешних усилий.

При составлении мысленно отсеченных частей, условие равновесия тела обеспечивается соотношением:

Так как исходная система внешних сил= (1) эквивалентна нулю, получаем:

{S'} =3D -{S"}

(3)

Это условие соответствует четвертой аксиоме статики о равенстве сил действия и противодействия.

Используя общую методологию теоремы = Пуансо о приведении произвольной системы сил к заданному центру и выбрав за полюс приведения центр масс, сечения А',= точку С', систему внутренних усилий для левой части {S'} сводим к главному вектору и главному моменту внутренних усилий. Аналогично делается для правой отсеченной част= и, где положение центра масс сечения А" определяется, соответствен= но, точкой С" (Рис.1 б,в).

{S'} ~ {R',L'0};      {S"} ~ {R",L"0}

(4)

Здесь в соответствие с четвертой аксиомой статики по-прежнему имеют место следующие соотношения:

R' =3D -R"

(5)

L'0= =3D -L"0

Таким образом главный вектор и главный момент системы внутренних усилий, возникающие в ле= вой, условно отсеченной части бруса, равны по величине и противоположны по направлению главному вектору и главному моменту системы внутренних усилий, возникающих в правой условно отсеченной части.

График (эпюра) распределения численн= ых значений главного вектора и главного момента вдоль продольной оси бруса и предопределяют, прежде всего, конкретные вопросы прочности, жесткости и надежности конструкций.

Определим механизм формирования компонент внутренних усилий, которые характеризуют простые виды сопротивлен= ий: растяжение-сжатие, сдвиг, кручение и изгиб.

В центрах масс исследуемых сечений С' или С" зададимся соответственно левой (с', х', у', z') или правой (с", х"= ;, у", z") системами координатных осей (рис.1 в), которые в отличие от базовой системы координат x, у, z будем называть "следящими". Термин обусловлен их функциональным назначением. А именно: отслеживание изменения положения сечения А (рис.1 а) при условном смещении его вдоль продольной оси бруса, например при: = 0 ≤ х'1 ≤ а, а ≤ x'2 ≤ b и т.д., где 0, а и b - линейные размеры границ исследуем= ых участков бруса.

Зададимся положительными направления= ми проекций главного вектора или и главного момента или на координатные оси следящей системы (рис.1 б, в):

{N', Q'y, Q'z},     {M'x, M'y, M'z}

(6)

{N", Q"y, Q"z},  &= nbsp;  {M"x, M"y, M"z}

При этом положительные направления проекций главного вектора и главного момента внутренних усилий на оси следя= щей системы координат соответствуют правилам статики в теоретической механике: = для силы - вдоль положительного направления оси, для момента - против часовой стрелки при наблюдении со стороны конца оси. Они классифицируются следующим образом:

Nx - нормальная сила, при= знак центрального растяжения или сжатия;

Мx - внутренний крутящий момент, возникает при кручении;

Qz, Q= у - поперечные или перерезывающие силы - признак сдвиговых деформаций,

Му, М= z - внутренние изгибающие моменты, соответствуют изгибу.

Соединение левой и правой мысленно отсеченных частей бруса приводит к известному (3) принципу равенства по модулю и противоположной направленности всех одноимен= ных компонент внутренних усилий, а условие равновесии бруса определяется в виде= :

{P1, P2, P3, ... , N', N", Q'y, Q"y, Q'z, Q"z, M'x, M"x,

M'y, M"y, M'= z, M"z, ... , Pn-1, Pn} ~ 0

(7)

С учетом эквивалентности нулю исходн= ой системы сил (1) имеет место:

{N', N&q= uot;, Q'y, Q"y, Q'z, Q"z, М'x, M"x, M'y, M"y, М'z, M"z}~0

(8)

Как естественное следствие из соотношений 3,4,5 полученное условие является необходимым для того, чтобы одноименные компоне= нты внутренних усилий попарно образовали подсистемы сил эквивалентные нулю:

  1. {N', N"} ~ 0 > N' =3D -N"
  2. {Q'<= sub>y, Q"y} ~ 0 > Q'y =3D -Q"y =
  3. {Q'<= sub>z, Q"z} ~ 0 > Q'z =3D -Q"z =
  4. {М'<= sub>x, M"x} ~ 0 > М'x =3D -M"x =
  5. {M'<= sub>y, M"y} ~ 0 > M'y =3D -M"y =
  6. {М'z, M"z} ~ 0= > М'z =3D -M"z

(9)

Общее число внутренних усилий (шесть= ) в статически определимых задачах совпадает с количеством уравнений равновесия= для пространственной системы сил и связано с числом возможных взаимных перемеще= ний одной условно отсеченной части тела по отношению к другой. Эти перемещения могут наблюдаться при разрушении тела по этому сечению.

Искомые усилия определяются из соотв= етствующих уравнений для любой из отсеченных частей в следящей системе координатных ос= ей. Так, для любой отсеченной части соответствующие уравнения равновесия приобретают вид;

  1. ix= =3D N + P1x= + P2x + ... + Pkx =3D 0 ® N
  2. iy= =3D Qy + P<= sub>1y + P2y + … + Pky =3D 0 ® = Qy
  3. iz =3D Q + P1z + P2z + ... + Pkz =3D 0 ® Qz
  4. x= (Pi) =3D Mx + Mx(Pi) + ... + Mx(Pk) =3D 0 ® M= x
  5. y= (Pi) =3D My + My(Pi) + ... + My(Pk) =3D 0 ® M= y
  6. z= (Pi) =3D Mz + Mz(Pi) + ... + Mz(Pk) =3D 0 ® M= z

(10)

Здесь для простоты обозначений систе= мы координат с' х' у' z' и с" х" у" т" замен= ены единой оxуz.

Уважаемые коллеги! Таким обра= зом, механизм предложенного автором лекций метода построения эпюр внутренних уси= лий, освобождающий Вас от механического запоминания "правил знаков" при построении эпюр внутренних усилий, заключается в следующем:

  1. Определите реакции в связях по величине и направлению в базовой системе координат.
  2. Опре= делите количество участков бруса для использования метода сечений.
  3. Мысл= енно рассеките брус в пределах исследуемого участка и изобразите на Ваше усмотрение левую или правую условно отсеченную часть.
  4. Укаж= ите пределы изменения положения сечения вдоль продольной оси в базовой сис= теме координат на этом участке.
  5. Введ= ите в искомом сечении соответственно левую или правую следящую систему координатных осей.
  6. Зада= йтесь положительными направлениями внутренних усилий в следящей системе координат.
  7. Сост= авьте уравнения равновесия для рассматриваемой условно отсеченной части брус= а в следящей системе координат.
  8. Опре= делите из уравнений равновесия искомые внутренние усилия.
  9. Вычи= слите искомые внутренние усилия на границах участков и при необходимости, - = их экстремальные значения.
  10. Выбр= ав масштаб усилий, выполните построение эпюры в соответствие с полученным= и их модульными значениями и знаками.

Указанная последовательность действи= й (кроме п.1) составляет суть метода сечений (разреза), единственного метода для определения внутренних усилий.

Не забываем, что при распределенной нагрузке в соответствие с теоремой Вариньона векторный момент равнодействующей рассматриваемой системы сил относительно любой точки равен сумме векторных моментов всех сил этой системы относительно той же точки.

Эпюры внутренних усилий позволяет визуально найти положение опасного сечения, где действуют наибольшие по мод= улю внутренние усилия. В этом сечении при прочих равных условиях наиболее вероя= тно разрушение конструкции при предельных нагрузках.

Эпюры внутренних усилий при растяжении-сжатии и кручении

Ключевые слова: Нормальное сечение. Нормальная сила. Внутренний крутящий момент.

Эпюры внутренних усилий при растяжении-сжатии

Растяжением или сжатием называется такой простой вид сопротивления, при котором внешние силы приложены вдоль продольной оси бруса, а в поперечном сечении его возникает только нормальная сила.

Рассмотрим расчетную схему бруса постоянного поперечного сечения с заданной внешней сосредоточенной нагрузко= й Р и распределенной q, (рис.1).

Пусть . Прежде всего определим опорную реакц= ию R, задавшись ее направлением вдоль оси х.

Брус имеет 2 участка и .

В пределах первого участка мысленно рассечем брус на 2 части нормальным сечением и рассмотрим равновесие, допус= тим левой части, введя следующую координату х1, рис.1 б:

Следовательно, в пределах первого участка брус претерпевает сжатие постоянной нормальной<= /span> силой.

Аналогично поступим со вторым участк= ом. Мысленно рассечем его сечением 2-2, и рассмотрим равновесие левой части (рис.1 в).Установим предварительно границы изменени= я х2:

Подставляя граничные значения параме= тра х2, получим:

Таким образом, в пределах второго участка брус растянут и нормальная сила изменяет= ся по линейному закону.

Аналогичный результат получается и п= ри рассмотрении правой отсеченной части (рис.1 г):

На основе полученных данных строится эпюра нормальных сил в виде графика распределения нормальной силы по длине бруса (рис.1 д). Характерно, что скачки на эпюре обусловл= ены наличием в соответствующих сечениях сосредоточенных сил R и Р= .

Эпюры внутренних усилий при кручении

Кручением называется простой вид сопротивления, при котором к брусу (валу) прикладыва= ются внешние пары сил в плоскостях, совпадающих с поперечным сечением вала, а в последних возникает только внутренний крутящий момент.

Рассмотрим расчетную схему вала, нагруженного двумя сосредоточенными моментами М и 2М и распределенными по д= лине: m, рис.2.

Методика построения эпюры аналогична только что рассмотренной методике при растяжении-сжатии.

В исходных сечениях № 1,2 и 3 задают= ся положительными значениями внутренних крутящих моментов М1, М2, М3. Пусть М=3Dml.

Для первого участка (рис.2 б):

Для второго участка (рис.2 в):

Для третьего участка (рис.2 г):

Границы измерения параметра х3 в следующей системе координат:

Тогда:

Отмеченные значения ординат откладываются на эпюре внутренних крутящих моментов (рис.2 д).

Эпюры внутренних усилий при прямом изгибе.

Ключевые слова: поперечная сила. Внутренний изгибающий момент.

Прямым изгибом называется такой в= ид простого сопротивления, когда внешние силы приложены перпендикулярно продол= ьной оси бруса (балки) и расположены в одной из главных плоскостей в соответстви= е с конфигурацией поперечного сечения балки.

Как известно, при прямом изгибе в поперечном сечении возникают два вида внутренних усилий: поперечная сила и внутренний изгибающий момент.

Рассмотрим пример расчетной схемы консольной балки с сосредоточенной силой Р, рис. 1, а, но…

Предварительно рекомендую Вам вспомнить из раздела "Статика" теоретической механики методы расч= ета реакций в связях на примерах тестов, приведенных в ПРИЛОЖЕНИИ по разделом Т-2.=

Прежде всего вычислим реакции в связи на базе уравнений равновесия:

После мысленного рассечения балки нормальным сечением 1-1 рассмотрим равновесие левой отсеченной части (рис.1, б), получим:

Таким образом, на первом участке поперечная сила отрицательная и постоянная, а внутренний изгибающий момент изменяется по линейному закону.

Для правой отсеченной части при рассмотрении ее равновесия результат аналогичен рис.1, в. А именно:

На основании полученных значений строятся эпюры поперечных сил (рис.1, г) и внутренних изгибающих моментов (рис.1, д).

Как следует из построенных эпюр , а в сечении жесткой связи. Именно это сечение и является наиболее опасным в данной расчетной схеме.

Продифференцируем выражение внутренн= его изгибающего момента по координате х:

Как видим, после дифференцирования получено выражение для поперечной силы. Случайность это или закономерность?= - Закономерность.

Дифференциальные зависимости между внутренними усилиями при изгибе

Рассмотрим расчетную схему балки с произвольной распределенной нагрузкой (рис.2).

Составим уравнение равновесия:

Таким образом, действительно: первая производная от внутреннего изгибающего момента по линейной координате равна поперечной силе в сечении.

Это известное свойство функции и ее первой производной успешно используется при пр= оверке правильности построения эпюр. Так, для расчетной схемы консольной балки (рис.1) эта связь дает следующие проверочные результаты:

и М убывает от 0 до -Pl.

и М º х<= /i>.

Таким образом, для квалифицирован= ной проверки Вам рекомендуется вспомнить из высшей математики раздел, связанный= с вычислением производных функции. Считаю целесообразно решить тесты, приведе= нные в ПРИЛОЖЕНИИ под разделом Т-3.=

Рассмотрим ВТОРОЙ ХАРАКТЕРНЫЙ ПРИМЕР ИЗГИБА двухопорной балки (рис.3).

Очевидно, что опорные реакции RA = =3D RB :

для первого участка (рис.3, б) -

для второго участка (рис.3, в) -

Эпюры внутренних усилий представлены соответственно на рис.3, г и 3, д.

На основе дифференциальной связи = Q и М, получим:

  • для первого участка:

Q > 0 и М возрастает от нуля до .

Q =3D const и M º x

  • для второго участка:

Q < 0 и М убыва= ет с до нуля.

Q =3D const и M также = пропорционален х, т.е. изменяется по линейному закону.

Опасным в данном примере является сечение балки в центре пролета:

ТРЕТИЙ ХАРАКТЕРНЫЙ ПРИМЕР

связан с использованием распределенн= ой по длине балки нагрузки (рис.4). Следуя методике, принятой ранее, очевидно равенство опорных реакций: , а для искомого сечения (рис.4, б) выражения для внутренних усилий приобретают вид:

На обеих опорах изгибающий момент отсутствует. Тем не менее опасным сечением балки= будет центр пролета при . Действительно, исходя из свойства функции и производной при , внутренний изгибающий момент достигает экстремума. Для нахожден= ия исходной координаты х0 (рис.3 в) в общем случае приравняем выражение поперечной силы к нулю. В итоге получим

После подстановки в выражение изгибающего момента получим:

Таким образом, .

Необходимо отметить, что техника построения эпюр при изгибе наиболее трудно усваивается слушателями. Вам представляется возможность научиться "быстрому" построению эпюр на тесторе-тренажере, приведенном в ПРИЛОЖЕНИИ под грифом Т-4.=

Понятие о напряжениях и деформациях

Ключевые слова: нормальное и касательное напряжения, линейная и угловая деформации, тензор напряжений.

Как отмечалось выше, внутренние силы, действующие в некотором сечении со стороны отброшенной части тела, можно привести к главному вектору и главному моменту. Зафиксируем точку М в рассматриваемом сечении с единичным вектором нормали n. В окрестности этой точки выделим малую площадку DF. Главный вектор внутренних сил, действующих на этой площадке, обозначим чере= з DP (рис. 1, а). При уменьшении размеров площадки соответственно

уменьшаются главный векто= р и главный момент внутренних сил, причем главный момент уменьшается в большей степени. В пределе при DF<= span style=3D'font-family:Symbol'>®0 получим

Аналогичный предел для главного моме= нта равен нулю. Введенный таким образом вектор рn называется вектором напряжений в точке. Этот вектор зависит не тольк= о от действующих на тело внешних сил и координат рассматриваемой точки, но и от ориентации в пространстве площадки DF, характеризуемой вектором n. Совокупность всех векторов напряжений в точке М для всевозможных направлений вектора n определяет напряженное состояние в этой точке.

В общем случае направление вектора напряжений рn не совпадает с направле= нием вектора нормали n. Проекция вектора рn на направление вектора n называется нормальным напряжением sn, а проекция на плоскость, проходящую через точку М и ортогональную вектору n, - касательным напряжением t<= sub>n (рис. 1 б).

Размерность напряжений равна отношен= ию размерности силы к размерности площади. В международной системе единиц СИ напряжения измеряются в паскалях: 1 Па=3D1 Н/м2.

При действии внешних сил наряду с возникновением напряжений происходит изменение объема тела и его формы, т. = е. тело деформируется. При этом различают начальное (недеформированное) и коне= чное (деформированное) состояния тела.

Отнесем недеформированное тело к декартовой системе координат Oxyz (рис. 2). Положение некоторой точки М в этой системе координат определяется радиус-вектором r(х, у, z). В деформированном состоянии точка М займет новое положение М', характеризуемое радиус-вектором r' (х, у, z). Вектор u=3Dr'-r называется вектором перемещений точки М. Проекции вектора u на координатные оси определяют компоненты вектора перемещений u(х, у, z), v(х, у, z), w(х, у, z), равные разности декартовых координат точки = тела после и до деформации.

Перемещение, при котором взаимное расположение точек тела не меняется, не сопровождается деформациями. В этом случае говорят, что тело перемещается как жесткое целое (линейное перемещен= ие в пространстве или поворот относительно некоторой точки). С другой стороны, деформация, связанная с изменением формы тела и его объема, невозможна без перемещения его точек.

Деформации тела характеризуются изменением взаимного расположения точек тела до = и после деформации. Рассмотрим, например, точку М и близкую к ней точку N= , расстояние между которыми в недеформированном состоянии вдоль направления вектора s обозначим через Ds (рис. 2). В деформированном состоянии точки М и N переместятся в новое положение (точки М' и N'), расстояние между которыми обозначим через Ds'. Предел отношения

называется относительной линейной деформацией в точке М в направлении вектора s. Рассматрив= ая три взаимно перпендикулярных направления, например, вдоль координатных осей= Ох, Оу и Oz, получим три компоне= нты относительных линейных деформаций ex, ey, ez, характеризующих измене= ние объема тела в процессе деформации.

Для описания деформаций, связанных с изменением формы тела, рассмотрим точку М и две близкие к ней точки = N и Р, расположенные в недеформированном состоянии в направлении двух взаимно ортогональных векторов s1 и= s2. Расстояния между точками обозначим через Ds1 и Ds2 (рис. 4). В деформированном состоянии положение точек обозначим через М', = N' и Р'. Угол между отрезками M'N'= и М'Р' в общем случае будет отличным от пря= мого. При Ds1®0, = Ds2®0 изменение угла g12 между двумя ортогональными до деформации направлениями называется угловой деформацией. Как видно из рис. 4, угловая деформация складывается из двух углов a1 и a<= sub>2, связанных с поворотами отрезков M'N' и М'Р' в плоскости, образованной векторами s1 и s2, относите= льно этих векторов. Если заданы три взаимно ортогональных вектора, направленных вдоль координатных осей, то имеются три угловые деформации gxy, gxz и g= yz, которые вместе с тремя линейными деформациями ex, ey и ez полностью определяют деформированное состояние в точке.

Напряженное состояние в точке. Тензор напряжений

Вектор напряжений pn явля= ется физическим объектом, имеющим длину, направление и точку приложения. В этом = смысле он обладает векторными свойствами. Однако этому объекту присущи некоторые свойства, не характерные для векторов. В частности, величина и направление вектора напряжений зависят от ориентации вектора n нормали бесконечно малого элемента поверхности dF. Совокупность всех возможных пар вект= оров n, рn в точке определяет напря= женное состояние в данной точке. Однако для полного описания напряженного состояни= я в точке нет необходимости задавать бесконечное множество направлений вектора = n, достаточно определить векторы напряжений на трех взаимно перпендикулярных элементарных площадках. Напряжения на произвольно ориентированных площадках могут быть выражены через эти три вектора напряжений. В дальнейшем лектор умышленно меняет ориентацию координат. Так, что ось Z - продольная о= сь бруса, а X и Y - координаты любой точки его поперечного сечен= ия.

Проведем через точку М три взаимно перпендикулярных плоскости с векторами нормалей, направления которых совпадают с направлениями координатных осей. Элементарные площадки образуем дополнительными сечениями, параллельными исходным плоскостям и отстоящими от них на бесконечно малые расстояния dx, dy, dz. В результате в окрестности точки М получим бесконечно малый параллелепипед, поверхн= ость которого образована элементарными площадками dF= х=3Ddydz, dFн=3Ddxdz, dFя=3Ddxdy. В= екторы напряжений px, py, pz, действующие на элементарных площадках, показа= ны на рис. 5.

Разложим каждый вектор напряжений на составляющие вдоль координатных осей (рис. 6). На каждой площадке действует одно нормальное напряжение <= span style=3D'font-family:Symbol'>sx, sy, sz, где индекс обозначает направление вектора нормали к площадке и два касат= ельных напряжения t с двумя индексам= и, из которых первый указывает направление действия компоненты напряжения, второй-направление вектора нормали к площадке.

Совокупность девяти компонент напряж= ений (по три на каждой из трех взаимно перпендикулярных площадок) представляет с= обой некоторый физический объект, называемый тензором напряжений в точке. Тензор можно представить в виде матрицы, соответствующим образом упорядочив девять компонент:

Для компонент тензора напряжений общепринятым является следующее правило знаков: компонента считается положительной, если на площадке с положительной внешней нормалью (т. е. направленной вдоль одной из координатных осей) э= та компонента направлена в сторону положительного направления соответствующей = оси. На рис. 6 все компоненты тензора напряжений изображены положительными. На площа= дках с отрицательной внешней нормалью (грани параллелепипеда, не видимые на рис. 5 и 6) положительная компонента направлена в противоположном направлени= и. Напряжения на трех взаимно ортогональных площадках с отрицательными направлениями нормалей также характеризуют напряженное состояние в точке. Э= ти напряжения, являющиеся компонентами тензора напряжений = , определяются аналогично напряжениям на площадках с положительной нормалью. = Они обозначаются теми же символами и имеют положительное направление, обратное = изображенному на рис. 6.

Свойства тензора напряжений. Главные напряжения

Ключевые слова: шаровый те= нзор напряжений, инвариантность, характеристическое уравнение, девиатор.

Тензор напряжений обладает свойством симметрии. Для доказательства этого свойства рассмотрим приведенный в лекци= и 5 элементарный параллелепипед с действующими на его площадках компонентами тензора напряжений. Так как тело находится в равновесии, следовательно, находится в равновесии любая его часть, в том числе и э= лементарный объем. Запишем одно из шести уравнений равновесия этого объема, а имен но - сумму моментов всех сил относительно оси Ох. Все силы, кроме двух, либо не создают момента относительно ocи Ох, либо взаимно уничтожаются. Отличные= от нуля моменты создают компоненты t= yz (верхняя грань) и tzy (правая грань):

После сокращения на элемент объема <= i>dV=3Ddxdydz получим

Аналогично, приравнивая нулю сумму моментов всех сил относительно осей Оу и Оz, получим еще два соотношения

Эти условия симметрии и тензора напряжений называются также условиями парности касательных напряжений: касательные напряжения, действующие по двум взаимно перпендикулярным площад= кам в направлениях, ортогональных ребру, образованному пересечением этих площад= ок, равны по величине. С учетом этих свойств из девяти компонент тензора напряж= ений независимыми оказываются шесть компонент.

Покажем теперь, что компоненты тензо= ра напряжений определенные для трех взаимно перпендикулярных площадок, полност= ью характеризуют напряженное состояние в точке, т. е. позволяют вычислить компоненты вектора напряжений на площадках, произвольно ориентированных относительно выбранной системы координат. Для этого рассмотрим элементарный объем, образованный сечением параллелепипеда, изображенного на рис. 1, плоскостью, пересекающей координатные оси и имеющей единичный вектор нормали

n с компонентами <= i>nx, ny, nz. На гранях полученного таким образом бесконечно малого тетраэдра действуют напряжения, показанные на рис. 1. При этом вектор напряжений pn на наклонной площадке разложен па составляющие рx, рy, рz вдоль координатных осей. Площади граней, ортогональных координатным осям и вектору нормали, обозначим соответственно dFx, dFy, dFz, dF. Эти площади связаны между собой соотношениями

dFx=3DdFnx, dFy=3DdFny, dFz=3DdFnz

(1)

вытекающими из того, что грани, ортогональные координатным осям, есть проекции наклонной площадки на соответствующую координатную плоскость.

Проектируя силы, действующие на гран= ях элементарного тетраэдра, на координатные оси, получим уравнения равновесия = для рассматриваемого объема. Например, проекции всех поверхностных сил на ось Ох дают

С учетом соотношений (1) после сокращения на dF получим уравнение, связывающее проекцию р<= span class=3DGramE>x вектора напряжений с соответствующими компонентами тензора напряжений. Объединяя это уравнение с двумя аналогичны= ми уравнениями, полученными проектированием сил на оси Оy и Оz, приходим к следующим соотношениям

(2)

носящим название формул Коши.= Эти формулы определяют вектор напряжений на произвольно выбранной площадке с вектором n через компоненты тензора напряжений.

Формулы (2) позволяют вычислить через компоненты тензора напряжений

полное напряжение

(3)

нормальное напряжение

(4)

и касательное напряжение:

Среди всех возможных направлений век= тора нормали n существуют такие направления, для которых вектор напряжени= й pn параллелен вектору n. На соответствующих площадках действуют только нормальные напряжения, а касательные напряжения отсутствуют. Такие площа= дки называются главными, а нормальные напряжения на этих площадках называются главными напряжениями. Пусть площадка с единичным вектором нормали является главной. Условия коллинеарности векторов pn<= /i> и n есть условия пропорциональности их компонент:

С учетом формул Коши получим систему= линейных однородных уравнений относительно неизвестных компонент nx, ny, nz вектора нормал= и к главной площадке

Эта система уравнений имеет ненулевое решение, если определитель, составленный из коэффициентов уравнений, обраща= ется в нуль:

Раскрывая определитель, приходим к кубическому уравнению относительно главного напряжения s

Здесь введены обозначения

Уравнение (3) называется характеристическим уравнением для тензора напряжений. Коэффициенты (4) этого уравнения называются инвариантами тензора напряжений. Решение кубического уравнения (3) имеет три вещественных корня s1, s2, s3= , которые обычно упорядочиваются s1 ³ s2 ³ s3.

Каждому значению sj (j=3D1, 2, 3) соответствует вектор n= j, характеризующий положение j-й главной площадки, с компонентами n<= sup>j1, nj2, nj3. Для нахождения эт= их компонент достаточно в уравнения подставить найденное значение sj и решить любые два из этих уравнений совместно с условием нормировки

(5)

Главные напряжения обладают важным свойством: по сравнению со всеми другими площадками нормальные напряжения на главных площадках принимают экстремальные значения. Для доказательства этого свойства достаточно исследовать на экстремум нормальное напряжение как функ= цию nx, ny, nz при дополнительном ограничении (5). Можно показать, что три главные площадки, соответствующие главным напряжени= ям s1, s2, s3, взаимно перпендикулярны или, что то же самое, векторы nj и nk, соответствующие различным значениям j и k - ортогональны. Усл= овие ортогональности имеет вид

Кубическое уравнение (3) можно переписать в виде

Приводя это уравнение к виду (3), получим следующие выражения для инвариантов (4) через главные напряжения:

Термин "инвариантность" обозначает независимость некоторой величины от выбора системы координат.

Введем среднее (гидростатическое) напряжение по формуле

Тензор напряжений можно представить в виде суммы двух тензоров , где

Первый тензор называется шаровым<= /i>, он характеризует изменение объема тела без изменения его формы. Второй тенз= ор, называемый девиатором, характеризует изменение формы. Особенностью девиатора напряжений является равенство нулю его первого инварианта:

Найдем положение площадок, на которых касательные напряжения принимают экстремальные значения. Для этого нужно отыскать экстремумы касательного напряжения при ограничении (5). Экстремальные касательные напряжения действуют на площадках, параллельных одной из главных осей и образующих с двумя другими осями угол = p/4. По величине эти напряжения равны

При этом на площадках с экспериментальными касательными напряжениями присутствуют нормальные напряжения, которые равны

Фигура, которую образуют площадки с экстремальными касательными напряжениями, изображена на рис. 2. Она принадлежит к классу параллелоэдров и представляет собой 12-гран= ник с гранями в виде ромбов, отношение диагоналей которых равно .

Таким образом, общая теория напряжен= ного состояния позволяет охватывать, в целом, весь комплекс видов сопротивлений,= как простого, так и сложного характера.

Плоское напряженное состояние

Ключевые слова: экстремаль= ные напряжения, тензор деформации.

Рассмотрим важный для приложений слу= чай плоского напряженного состояния, реализуемого, например, в плоскости Oyz= . Тензор напряжений в этом случае имеет вид

Геометрическая иллюстрация представл= ена на рис.1. При этом площадки х=3Dconst являются главными с соответствующими нулевыми главными напряжениями. Инварианты тензора напряжений равны , а характеристическое уравнение принимает вид

Корни этого уравнения равны

(1)

Нумерация корней произведена для слу= чая s1>0, s1<0.

Произвольная площадка характеризуется углом a на рис. 1, при этом вектор n имеет компоненты: ny=3Dcosa, nz=3Dsina, nх=3D0. Нормальное и касательное напряжения на наклонной площадке выражаются через угол a следующим образом:

(2)

(3)

Так как на главных площадках касател= ьное напряжение отсутствует, то, приравнивая нулю выражение (3), получим уравнение для определения угла a между нормалью n и осью = Оу

(4)

Наименьший положительный корень уравнения (4) обозначим через a1. Та= к как tg(х)-периоди= ческая функция с периодом p, то имеем два взаимно ортогональных направления, составляющие углы a1 и a<= sub>2=3Da1 + p/2 с осью Оу. Эти направления соответствуют взаимно перпендикулярным главным площадкам (рис. 2).

Если продифференцировать соотношение= (2) по a и приравнять производную нул= ю, то придем к уравнению (4), что доказывает экстремальность главных напряжений.

Для нахождения ориентации площадок с экстремальными касательными напряжениями приравняем нулю производную от выражения

откуда получим

(5)

Сравнивая соотношения (4) и (5), находим, что

Это равенство возможно, если углы 2<= span style=3D'font-family:Symbol'>a и 2at отличаются на угол p/2. Следовате= льно, направления площадок с экстремальными касательными напряжениями отличаются от направлений главных площадок на угол p/4 (рис. 3).

Величины экстремальных касательных напряжений получим после подстановки (5) в соотношение (3) с использованием формул

После некоторых преобразований получ= им

Сравнивая это выражение с полученными ранее значениями главных напряжений (1), выразим экстремальные касательные напряжения через главные напряжения

Аналогичная подстановка в (2) приводит к выражению для нормальных напряжений на площадках с at

Полученные соотношения позволяют проводить направленно-ориентированный расчет конструкций на прочность в слу= чае плоского напряженного состояния.

Тетзор деформации

Рассмотрим вначале случай плоской деформации (рис. 4). Пусть плоский элемент MNPQ перемещается в пределах плоскости и деформируется (изменяет форму и размеры). Координаты точек элемента до и по= сле деформации отмечены на рисунке.

По определению относительная линейная деформация в точке М в направлении оси О<= /span>х равна

Из рис. 4 следует

Учитывая, что MN=3Ddx, получи= м

В случае малых деформаций, когда (дu/дх)&= lt;<1, (дv/дх)<< 1, можно пренеб= речь квадратичными слагаемыми. С учетом приближенного соотношения

справедливого при x<<1, окончательно для малой деформации получим

Угловая деформация gxy определяется как = сумма углов a1 и a2 (4). В случае малых деформаций

Для угловой деформации gxy имеем

Проводя аналогичные выкладки в общем случае трехмерной деформации, имеем девять соотношений

(6)

связывающих линейные и угловые деформации с перемещениями. Эти соотношения носят назван= ие соотношений Коши.

Три линейных и шесть угловых деформа= ций (6) образуют тензор малых деформаций

(7)

Этот тензор полностью определяет деформированное состояние твердого тела. Он обладает теми же свойствами, чт= о и тензор напряжений. Свойство симметрии непосредственно следует из определения угловых деформаций. Главные значения и главные направления, а также экстремальные значения угловых деформаций и соответствующие им направления находятся теми же методами, что и для тензора напряжений.

Инварианты тензора деформаций определяются аналогичными формулами, причем первый инвариант тензора малых деформаций имеет ясный физический смысл. До деформации его объем равен d= V0=3Ddxdydz. Если пренебречь деформациями сдвига, которые изменяют форму, а не объем, то после деформации ребра будут иметь размеры

(рис. 4), а его объем будет равен

Относительное изменение объема

в пределах малых деформаций составит=

что совпадает с определением первого инварианта. Очевидно, что изменение объема есть физическая величина, не зависящая от выбора системы координат.

Так же, как и тензор напряжений, тен= зор деформаций можно разложить на шаровой тензор и девиатор. При этом первый инвариант девиатора равен нулю, т. е. девиатор характеризует деформацию тела без изменения его объема.

Упругость и пластичность. Закон Гука

Ключевые с= лова: упругость, пластичность, разрушение, коэффициент Пуассона, модуль Юнга, мод= уль сдвига, энергия деформации.

Действие внешних сил на твердое тело приводит к возникновению в точках его объема напряжений и деформаций. При э= том напряженное состояние в точке, связь между напряжениями на различных площад= ках, проходящих через эту точку, определяются уравнениями статики и не зависят от физических свойств материала. Деформированное состояние, связь между перемещениями и деформациями устанавливаются с привлечением геометрических = или кинематических соображений и также не зависят от свойств материала. Для того чтобы установить связь между напряжениями и деформациями, необходимо учитыв= ать реальные свойства материала и условия нагружения. Математические модели, описывающие соотношения между напряжениями и деформациями, разрабатываются = на основе экспериментальных данных. Эти модели должны с достаточной степенью точности отражать реальные свойства материалов и условия нагружения.

Наиболее распространенными для конструкционных материалов являются модели упругости и пластичности. Упр= угость-это свойство тела изменять форму и размеры под действием внешних нагрузок и восстанавливать исходную конфигурацию при снятии нагрузок. Математически свойство упругости выражается в установлении взаимно од= нозначной функциональной зависимости между компонентами тензора напряжений и тензора деформаций. Свойство упругости отражает не только свойства материалов, но и услов= ия нагружения. Для большинства конструкционных материалов свойство упругости проявляется при умеренных значениях внешних сил, приводящих к малым деформациям, и при малых скоростях нагружения, когда потери энергии за счет температурных эффектов пренебрежимо малы. Материал называется линейно-уп= ругим, если компоненты тензора напряжений и тензора деформаций связаны линейными соотношениями.

При высоких уровнях нагружения, когд= а в теле возникают значительные деформации, материал частично теряет упругие свойства: при разгрузке его первоначальные размеры и форма полностью не восстанавливаются, а при полном снятии внешних нагрузок фиксируются остаточ= ные деформации. В этом случае зависимость между напряжениями и деформациями перестает быть однозначной. Это свойство материала называется пластичностью= . Накапливаемые в процессе пластического деформирования остаточные деформации называются пластическими.

Высокий уровень нагружения может выз= вать разрушение, т. е. разделение тела на части. Твердые тела, выполненны= е из различных материалов, разрушаются при разной величине деформации. Разрушение носит хрупкий характер при малых деформациях и происходит, как правило, без заметных пластических деформаций. Такое разрушение характерно для чугуна, легированных сталей, бетона, стекла, керамики и некоторых других конструкционных материалов. Для малоуглеродистых сталей, цветных металлов, пластмасс характерен пластический тип разрушения при наличии значительных остаточных деформаций. Однако подразделение материалов по характеру разруше= ния на хрупкие и пластичные весьма условно, оно обычно относится к некоторым стандартным условиям эксплуатации. Один и тот же материал может вести себя в зависимости от условий (температура, характер нагружены я, технология 'изготовления и др.) как хрупкий или как пластичный. Например, пластичные п= ри нормальной температуре материалы разрушаются как хрупкие при низких температурах. Поэтому правильнее говорить не о хрупких и пластичных материа= лах, а о хрупком или пластическом состоянии материала.

Пусть материал является линейно-упру= гим и изотропным. Рассмотрим элементарный объем, находящийся в условиях одноосн= ого напряженного состояния (рис. 1), так что тензор напряжений имеет вид

При таком нагружении происходит увеличение размеров в направлении оси Ох, характеризуемое линейной деформацией , которая пропорциональна величине напряжения

(1)

Это соотношение является математичес= кой записью закона Гука, устанавливающего пропорциональную зависимость м= ежду напряжением и соответствующей линейной деформацией при одноосном напряженном состоянии. Коэффициент пропорциональности Е называется модулем продольной упругости или модулем Юнга. Он имеет размерность напряжений.

Наряду с увеличением размеров в направлении действия напряжения s= x происходит уменьшение размеров в двух ортогональных направлениях (рис. 1). Соответствующие деформации обозначим через ey(sx) и ez(sx), причем эти деформации отрицательны при положительных sx и пропорциональны ez:<= /p>

(2)

Коэффициент пропорциональности m называется коэффициентом Пуассона<= /i>, который в силу изотропности материала одинаков для обоих ортогональных направлений.

Соотношения, аналогичные (1) и (2), в случае одноосного нагружения в направлении осей Оу, Оx напряжением s= y, sz, соответственно име= ют вид

(3)

(4)

При одновременном действии напряжени= й по трем ортогональным осям, когда отсутствуют касательные напряжения, для линейно-упругого материала справедлив принцип суперпозиции (наложения решен= ий):

С учетом формул (1) - (4) получим

(5)

Касательные напряжения вызывают угло= вые деформации, причем при малых деформациях они не влияют на изменение линейных размеров, и следовательно, на линейные деформации. Поэтому они справедливы также в случае произвольного напряженного состояния и выражают = так называемый обобщенный закон Гука.

Угловая деформация gxy обусловлена касательным напряжением txy, а деформации gxz и gyz - соответственно напряже= ниями txz и tyz. Между соответствующими касательными напряжен= иями и угловыми деформациями для линейно-упругого изотропного тела существуют пропорциональные зависимости

(6)

которые выражают закон Гука при сдвиге. Коэффициент пропорциональности G<= /i> называется модулем сдвига. Существенно, что нормальное напряжение не влияет на угловые деформации, так как при этом изменяются только линейные размеры отрезков, а не углы между ними (рис. 1).

Линейная зависимость существует также между средним напряжением, пропорциональным первому инварианту тензора напряжений, и объемной деформацией, совпадающей с первым инвариантом тензора деформаций:

(7)

Соответствующий коэффициент пропорциональности К называется объемн= ым модулем упругости.

В формулы (1) - (7) входят упругие характеристики материала Е, m<= /span>, G и К, определяющие его упругие свойс= тва. Однако эти характеристики не являются независимыми. Для изотропного материа= ла независимыми упругими характеристиками являются две, в качестве которых обы= чно выбираются модуль упругости Е и коэффициент Пуассона m. Чтобы выразить модуль сдвига G через Е и m, рассмотрим пл= оскую деформацию сдвига под действием касательных напряжений t (рис. 2). Для упрощения выкладок используем квадратный элемент со стороной а. Вычислим главные напряжения s1 =3D t, s3 =3D <= i>t. Эти напряжения действуют на площадках, расположенных под углом p/4 к исходным площадкам. Из рис. 2 найдем связь между линейной деформацией e1 в направлении действия напряжения s1 и угловой деформацией g. Большая диагональ = ромба, характеризующая деформацию e= 1, равна

Для малых деформаций tg g g, С учетом этих соотношений

До деформации эта диагональ имела ра= змер АВ=3Dа . Тогда будем иметь

Из обобщенного закона Гука (5) получим

откуда

Сравнение полученной формулы с запис= ью закона Гука при сдвиге (6) дает

G=3DE/[2(1+m)]

(8)

Сложим три соотношения упругости (5)

(9)

В итоге получим

Сравнивая это выражение с объемным законом Гука (7), приходим к результату

Механические характеристики Е= , m, G и К<= /i> находятся после обработки экспериментальных данных испытаний образцов на различные виды нагрузок. Из физического смысла все эти характеристики не мо= гут быть отрицательными. Кроме того, из последнего выражения следует, что коэффициент Пуассона для изотропного материала не превышает значения 1/2. Т= аким образом, получаем следующие ограничения для упругих постоянных изотропного материала:

E>= ;0, G>0, K>0, 0£ m £= 1/2,

Предельное значение m®1/2 приводит к предельному значению К® ¥, что соответствует несжимаемому материалу (q ® 0 при s0 ¹ 0). В заключение выразим из соотношений упругости (5) напряжения через деформации. Запишем первое из соотношений (5) в виде

С использованием равенства (9) будем иметь

откуда

Аналогичные соотношения можно вывести для sх и sy. В результате получим

(10)

Здесь использовано соотношение (8) для модуля сдвига. Кроме того, введено обозначение

Потенциальная энергия упругой деформации

Рассмотрим вначале элементарный объе= м dV=3Ddxdydz в условиях одноосного напряженного состояния (рис. 1). Мысленно закрепим площадку х=3D0 (рис. 3). На противоположную площадку действует сила sxdydz. Эта сила совершает работу на перемещении exdx. При увеличении напряжения от нулевого уровня до значения sx соответствующая деформация в силу закона Гука также увеличивается от нуля до значения = ex, а работа пропорциональна заштрихованной на рис. 4 площади: dA=3D0,5sxexdV. Если пренебречь кинетической энергией и потерями, связанными с тепловыми, электромагнитными= и другими явлениями, то в силу закона сохранения энергии совершаемая работа перейдет в потенциальную энергию, накапливаемую в процессе деформирования: dA=3DdU=3D0,5sxexdV. Величина Ф=3DdU/dV называется удельной потенци= альной энергией деформации, имеющей смысл потенциальной энергии, накопленной в единице объема тела. В случае одноосного напряженного состояния

 

При одновременном действии напряжений sx, sy и sz на главных площад= ках (т. е. при отсутствии касательных напряжений) потенциальная энергия равна с= умме работ, совершаемых силами sx= dydz, sydxdz, szdxdy на соответству= ющих перемещениях exdx, eydy, ezdz. Удельная потенциальная энергия равна

(2.47)

В частном случае чистого сдвига в плоскости Оху, изображенном на рис. 5, сила txydxdz= совершает работу на перемещении g= xydy. Соответствующая этому случаю удельная потенциальная энергия деформации равн= а

Подобные соотношения будут иметь мес= то при сдвиге в других плоскостях.

В общем случае напряженно-деформированного состояния будем иметь

(11)

Если деформации выразить через напряжения с помощью соотношений упругости (5) и (6), то получим эквивалентную форму записи через компоненты тензора напряжений

(12)

Выразив напряжения через деформации с использованием соотношений (6) и (10), получим еще одну форму записи для Ф - чер= ез компоненты тензора деформаций

Еще одну форму записи для удельной потенциальной энергии деформации получим, разложив тензоры напряжений и деф= ормаций на шаровые тензоры и девиаторы. В результате (11) можно привести к одной из форм

(13)

Здесь введены обозначения для t - интенсивности касательных напряж= ений и g - интенсивности деформаций сдвига, которые выражаются через вторые инварианты J2= (ds) и J2(de) девиаторов тензора напряжений и тенз= ора деформаций следующим образом:

Первые слагаемые в (13) соответствуют произведению шаровых составляющих тензоров напряжений и деформаций, а вторые - произведению девиаторных составляющих. Так как шаров= ой тензор характеризует изменение объема, а девиатор - изменение формы, то соотношения (13) можно интерпретировать как разложение удельной потенциальной энергии на две составляющие: Ф=3DФ0 + Фф, где Ф0 соответствует изменению объема без изменения формы= , а Фф - изменению формы без изменения объема. Первая составляющая будет вычислять= ся через компоненты тензора напряжений следующим образом:

(14)

Удельную потенциальную энергию измен= ения формы проще найти не через интенсивность касательных напряжений, а как разн= ость Ф - Ф0. Вычитая (14) из (12), после преобразований получим

Механические характеристики конструкционных материалов

Ключевые с= лова: упругое состояние; пластичное состояние; пределы пропорциональности, упруго= сти, текучести, прочности.

Механические характеристики определя= ются следующими факторами:

  • веще= ством, его структурой и свойствами;
  • конс= труктивными особенностями элемента, т. е, размерами, формой, наличием концентратор= ов, состоянием поверхности;
  • усло= виями при нагружении: температурой, скоростью, повторяемостью нагрузки и др.=

Конструкционные материалы в процессе деформирования вплоть до разрушения ведут себя по разно= му. Пластичное поведение характеризуется существенным изменением формы и размер= ов, при этом к моменту разрушения развиваются значительные деформации, не исчезающие после снятия нагрузки. Такие материалы называют пластичными. При хрупком поведении разрушение наступает при весьма малых деформациях, и материалы с такими свойствами называют хрупкими. Однако одни и те же конструкционные материалы, находящиеся в различных условиях деформирования, ведут себя по разному: при одних условиях проявл= яют себя как пластичные материалы, при других - как хрупкие. В связи с этим, основные макромеханические характеристики материалов - упругость, пластично= сть, вязкость и др. правильнее относить не к их свойствам, а к состояниям матери= ала.

Механические состояния деформирунмых тел

В упругом состоянии деформации обратимы, и вся энергия, затраченная на деформирование, при разгрузке возвращается (диссипация энергии отсутствует). Для любого твердого тела про= цесс деформирования начинается с упругой деформации. Изотропное тело имеет две константы упругости - модуль упругости Е и коэффициент Пуассона m. Для анизотропных тел число упругих констант в общем случае равно 21. Из основных констант упругости можно полу= чить их производные - модуль сдвига G, модуль объемной реформации К и постоянную Ламе l.

Вязкое сопротивление - в некотором смысле противоположно упругому - работа внешних сил, уравновешенных силами вязкого сопротивления, полностью рассеивается в виде тепла. Вязкое сопротивление определяется величиной касательной силы, необходимой для поддержания ламинарного скольжения слоев,= или течения с определенной скоростью. Таким образом вязкость можно определить как сопротивление течению.

Представление о вязкоупругой деформа= ции дает поведение моделей, сочетающих свойства вязкости и упругости в такой последовательности: при нагружении тела в нем возникает мгновенная упругая деформация, подчиняющаяся закону Гука; далее при том же макс= имальном напряжении наблюдается вязкая деформация, подчиняющаяся закону Ньютона.

Наиболее распространенными в теории линейной вязко-упругости являются реологические = модели Максвелла и Фойгта, дающие связь между напряжениями и деформациями и скорос= тями их изменения:

- модель Максвелла,

 

- модель Фойгта,

тде h - коэффициент вязкости.

Пластическое состояние характеризуется наличием остаточных деформаций, фиксируемых после снятия внешних нагрузок. Объем тела при пластической деформации не изменяется; усл= овие постоянства объема записывается в виде , (эксперименты показывают, что изменение объема не превышает 0,5= %).

В случае, когда все напряжения изменяются пропорционально одной из составляющих, в процессе пластической деформации направления главных деформаций совпадают с направлениями главных нормальных напряжений, направления максимальных сдвигов - с направлениями максимальных касательных напряжений, а главные направления девиатора напряж= ений - с главными направлениями девиатора деформаций.

Одной из распространенных моделей поведения материала при упруго-пластических деформациях является модель пластичности, основанная на деформационной теории Генки-Ильюшина, описываем= ая уравнениями:

Здесь

- средняя деформация,

- среднее напряжение,

y= - безразмерный коэффициент, называемый параметром пластичности (с точностью= до множителя он совпадает с интенсивностью касательных напряжений). При y=3D1 эта модель описывает поведение уп= ругого материала.

Высокоэластическое состояние - наиболее характерно для полимеров; особенностями этого состояния являются большая изменяемость формы и деформирование без изменения объема. Для материалов, находящихся в высокоэластическом состоянии, наблюдается существенная зависимость их свойств от длительности и скорости нагружения, = температуры и т. д.

Состояние разрушения - состоя= ние, при котором за счет интенсивного развития трещин в материале тела начинается нарушение его сплошности и непрерывности. Физический процесс разрушения материала представляется в виде двух основных стадий-стадии рассеянных разрушений (зарождение и развитие микроскопических трещин) и стадии развития магистральной трещины. Очаги зарождения микротрещин распределены по всему объему материала, находящегося в однородном напряженном состоянии, достаточ= но равномерно. Относительная длительность первой и второй стадии разрушения зависит от свойств материала, характера напряженного состояния и условий нагружения.

Диаграммы упруго-пластического деформирования конструкционных материалов

Основным опытом для определения меха= нических характеристик конструкционных материалов является опыт на растяжение призматического образца центрально приложенной силой, направленной по продольной оси; при этом в средней части образца реализуется однородное напряженное состояние. Форма, размеры образца и методика проведения испытан= ий определяются соответствующими стандартами, например, ГОСТ 34643-81, ГОСТ 1497-73. По результатам испытаний строится зависимость между напряжениями и деформациями , которая называется диаграммой деформирования. Опыты на растяжен= ие образцов выявляют некоторые общие свойства конструкционных материалов-свойства упругости и пластичности. На рис. 1 показаны типичные кривые деформирования при растяжении образцов из материала сталь 30 и сталь 40Х.

Если напряжения не превышают sпц - предела пропорциональности (точка / на диаграмме), и зависимость между напряжен= иями и деформациями линейна, то она описывается законом Гука , где Е - модуль продольной упругости материала. Размернос= ть модуля упругости-Н/м2 (Паскаль). Значение модуля упругости Е на= кривой деформирования численно равно тангенсу угла наклона линейного участка: . Таким образом, величину Е можно рассматривать как характеристику упругого сопротивления или как характеристику интенсивности нарастания напряжения с увеличением деформации. Физический смысл коэффициен= та Е определяется как напряжение, необходимое для увеличения длины образца в два раза. Такое толкование довольно искусственно, поскольку величина упругого удлинения у большинства твердых тел редко дости= гает даже 1 %.

Напряжения, являющиеся верхней грани= цей проявления чисто упругих деформаций, соответствуют точке 2 диаграммы и называются пределом упругости sупр.

Точка 3 диаграммы характерна тем, что при достижении напряжениями величины s =3D st (st - предел текучести), дальнейшее удлинение образца (для малоуглеродистых сталей) происходит практически без увеличения нагрузки. Это явление носит название текучести, а участок диаграммы, расположенный непосредствен= но правее точки 3, называется площадкой текучести. При этом полированная поверхность образца мутнеет, докрывается ортогональной сеткой линий (линии Чернова-Людерса), расположенных под углом 45° к продольной оси образца по направлению плоскостей действия максимальных касательных напряжений.

У многих конструкционных материалов площадка текучести не выражена столь явно, как у малоуглеродистых сталей. Д= ля таких материалов вводится понятие условного предела текучести ss; это напряжение, которому соответствует остаточная (пластическая) деформация, равная s%. Обычно принимается s =3D 0,2%.

После площадки текучести для дальней= шего увеличения деформации необходимо увеличение растягивающей силы. Материал сн= ова проявляет способность сопротивляться деформации; участок за площадкой текуч= ести (до точки 4) называется участком упрочнения. Точка 4 соответствует максимальной нагрузке, выдерживаемой образцом. Соответствующее напряжение называется временным сопротивлением sв (или пределом прочности sпч). Дальне= йшая деформация образца происходит без увеличения или даже с уменьшением нагрузки вплоть до разрушения (точка 5). Точке 4 на диаграмме соответствует начало локального уменьшения размеров поперечного сечения образца, где, в основном, сосредоточивается вся последующая пластическая деформация.

Диаграмма, приведенная на рис.1, является диаграммой условных напряжений, условность состоит в том, что все = силы относились к F0 - первоначальной площади поперечного сечения образца; в действительности же при растяжении площадь поперечного сечения образца уменьшается. Если учитывать текущее значение площади поперечного сечения при определении напряжений, то получим диаграмму истинных напряжений (рис. 2).

Если в некоторый момент нагружения (точка А на рис. 1) прекратить нагружение и снять нагрузку, то разгрузка образца пойдет = по линии АВ, параллельной линейному участку диаграммы 0-1. При этом полная деформация в точке А равна:

где - упругая деформация, - пластическая (остаточная деформация). Это уравнение справедливо для любой точки диаграммы.

После того как материал испытал воздействие осевого усилия одного знака (например, растяжение) в области пластических деформаций (s>st) сопротивляемость этого материала пластической деформации при действии сил другого знака (сжатие) понижается. Это явление носит название эффекта Баушингера.

При растяжении образца происходит не только увеличение его длины, но и уменьшение размеров поперечного сечения, = т. е. в упругой области деформация в поперечном направлении , где e - деформация в продольном направлении, m - ко= эффициент Пуассона. Для изотропных материалов значения коэффициента Пуассона находятся в пределах 0 <m £ 0,5.

Таблица 1. Механические характеристики некоторых материалов

Материал<= o:p>

Характери= стика

Е, ГПа

st, МПа

sв, МПа

d, %

y, %

Сталь Ст.3

200<= /o:p>

240/240

450/-

26

50

Сталь 15<= o:p>

200<= /o:p>

210/210

350/-

28

55

Сталь 45<= o:p>

200<= /o:p>

340/340

610/-

24

45

Сталь ЗОХ= ГСА

200<= /o:p>

950/950

1200/-

13

50

Чугун СЧ1= 5-32

150<= /o:p>

-

150/640

0,6<= /o:p>

-

Медь прут= ковая

110<= /o:p>

250/250

320/-

15

45

Дюралюмин= Д16

75

240/240

420/-

18

-

Дельта-др= евесина

20

-

250/160

-

-

Текстолит=

30

75/115

127/168

1,5<= /o:p>

-

Примечание.= В знаменателе указана соответствующая характеристика при сжати.

Для сталей различных марок Е = =3D 195-206 ГПа, G =3D 79-89 ГПа, m =3D 0,23-0,31, для сплавов алюминия Е =3D 69-71 ГПа, G =3D 26-27 = ГПа, m =3D 0,30-0,33. Упругие свойства некот= орых материалов даны в табл. 1.

Характеристиками пластичности матери= ала являются относительное удлинение и относительное сужение при разр= ыве:

где l0, F0<= /i> - длина рабочей части образца и площадь поперечного сечения до деформации; = lк - длина рабочей части обра= зца после разрыва; F0 - конечная площадь поперечного сечения в шейке образца после разрыва.

По величине относительного удлинения= при разрыве проводится разделение состояния материалов на пластичное и хрупкое. Материалы, имеющие к моменту разр= ушения достаточно большие значения d (d&= gt;10%), относят к пластическим материалам; к хрупким относят материалы с относительным удлинением d < 3= %.

Оценка пластических свойств материала может быть проведена по такой характеристике, IKBK ударная вязкость = -

KC=3DA/F= ,

где А - работа, затрачиваемая на ударное разрушение образца, Дж (или НЧм), F= - площадь поперечного сечения образца в месте концентратора, м2 (или см2),

Работа А деформации при разрушении образца может быть определена по диаграмме растяжения . Так, если первоначальная длина образца l0, то работа деформации, совершаемая силой Р на перемещении u:

где uк - перемещение в момент, предшествующий разрушению. Тогда по зависимости и , находим

где - площадь диаграммы деформирования (работа деформации на единицу объема материала). Для сталей КС=3D50-100 Н м/см2. Материалы с ударной вязкостью КС < 30 Н м/см2 относят к числу хрупких.

Некоторые пластичные материалы в рай= оне площадки текучести обнаруживают особенность (например титан), называемую "зубом текучести"; для таких материалов вводит= ся понятие верхнего и нижнего предела текучести (sтв, sтн).

Экспериментальное изучение свойств материалов при сжатии проводится на коротких образцах с тем, чтобы исключить возможность искривления образца. Для пластичных материалов характер диаграм= мы при сжатии примерно до возникновения текучести такой же, как и при растяжении. В процессе деформации сжатия образец укорачивается; при этом размеры поперечного сечения увеличиваются. Из-за трения между опорными плит= ами нагружающего устройства и торцевыми поверхностями образца он принимает бочкообразную форму. Для ряда пластичных материалов обнаружить напряжение, аналогичное временному сопротивлению при растяжении, не удается, так как образец сплющивается.

Хрупкие материалы проявляют значител= ьно лучшую способность сопротивляться деформациям сжатия, чем деформациям растяжения; для них разрушающее напряжение при сжатии превышает предел прочности при растяжении в несколько раз. Разрушение хрупких материалов при сжатии происходит за счет образования трещин.

Влияние различных факторов на механические характеристики конструкционных материалов

Ключевые слова: ползучесть, релаксация, длительная прочность.

Зависимость механических характерист= ик конструкционных материалов от их химического состава, внешних условий и усл= овий нагружения весьма многообразна; отметим наиболее существенные, характерные = для типичных условий эксплуатации конструкций.

Влияние содержания углерода. Введение различных легирующих добавок в металлы позволяет значительно повыс= ить прочностные характеристики сплавов. На рис. 1 показано влияние процентного содержания углерода на механические свой= ства конструкционной стали. Как видно, с увеличением содержания углевода, времен= ное сопротивление повышается в несколько раз; однако при этом значительно ухудшаются пластические свойства; относительное удлинение d и относительное сужение y при разрыве уменьшаются.

Влияние температуры окружающей ср= еды. Повышенные температуры оказывают существенное влияние на такие механические характеристики конструкционных материалов, как ползучесть и длительная прочность. Ползучестью называют медленное непрерывное возрастание пластической (остаточной) деформации под воздействием постоянных нагрузок. = Длительной прочностью называется зависимость разрушающих напряжений (временного сопротивления) от длительности эксплуатации. Свойства ползучести и длительн= ой прочности проявляются у углеродистых сталей при Т >300°С, для легированных сталей при Т>350°С. для алюминиевых сплавов при Т>10= 0°С. Некоторые материалы проявляют эти свойства и при обычных температурах.

Мерой оценки ползучести материала является предел ползучести - напряжение, при котором пластическая деформация за определенный промежуток времени достигает заданной величины. В некоторых случаях сопротивление ползучести оценивается величиной скорости деформации по прошествии заданного времени. При обозначении предела ползуче= сти указывается величина деформации, время и температура испытаний. Например, д= ля жаропрочного сплава ХН77ТЮР при температуре 700°С за время 100 часов и деформации ползучести 0,2% предел ползучести составляет 4= 00 МПа: s0,2/100 (700) = =3D400 МПа.

Ползучесть сопровождается релакса= цией напряжений - самопроизвольным уменьшением напряжений с течением времени= при неизменной деформации. Скорость релаксации напряжений возрастает при повыше= нии температуры. Мерой скорости релаксации служит время релаксации - промежуток времени, в течение которого напряжение уменьшается по сравнению с начальным значением в е=3D2,718 раза.

Прочность материала при повышенных температурах оценивается пределом длительной прочности - напряжением, при котором материал разрушается не ранее заданного времени. При обозначении предела длительной прочности указывается продолжительность нагружения и температура испытания. Так, для сплава ХН77ТЮР при температуре 700°С и времени 1000 часов предел длительной прочности составляет sдл 100(700= )=3D=3D330 МПа. При кратковременных испытаниях для этого же сплава при температуре 700= °С пределы прочности и текучести соответственно равны:= sв=3D830 МПа, s0,2=3D560 МПа.

Влияние повышенных температур на характеристики прочности и пластичности можно проследить на рис. 2 и 3, где представлены осредненные результаты экспериментов д= ля 1-углеродистой стали, содержащей 0,15% углерода; 2-0,40% углерода, 3-хромистой стали. Прочность углеродистых сталей с повышением температуры до 650-700°С снижается почти в десять раз. Наиболее резкое снижение sв наблюдается для алюминиевых сплавов. Наибольшими значениями sв при высоких температурах обладают литые жаропрочные сп= лавы, содержащие 70-80% никеля. Снижение пределов текучести sт с повышением температуры происходит примерно т= ак же, как и снижение sв.= Для углеродистых сталей характерным является ухудшение пластических свойств (охрупчивание) при температурах около 300°С (кри= вая 2 на рис. 3).

Влияние температур на упругие свойства. Температурный коэффициент линейного расширения и температурный коэффициент модуля упругости связаны между собой соотношением

или

где r и m - постоянные, характеризующие параметры кристаллической решетки. На рис. 4 приведена зависимость безразмерного модуля упругости Е/Е0 некоторых конструкционных материалов = от температуры (E0 - модуль упругости материала при обычной температуре): 1 - нержавеющая сталь; 2 - алюминиевые сплавы, 3 - углеродист= ые стали, 4 - титановые сплавы.

Для сталей с повышением температуры испытаний с 25 до 450°С модули упругости Е и G уменьшаются на 20-40%, при этом, начиная с 300-400°С наблюдается расхождение между значениями модулей, определенными при статических и динамических испытаниях.

Изменение модулей упругости при малый колебаниях температуры (от -50 до +500С) незначи= тельно и им обычно пренебрегают.

Основные понятия теории надежности конструкций=

Ключевые слова: коэффициент запаса, вероятность, коэффициент однородности, нормативы.

Постановка задач теории надежности

Согласно ГОСТ 27.002-89 "Надежн= ость в технике. Термины и определения" надежность конструкции есть свойство сохранять во времени способность к выполнению требуемых функций в заданных режимах. Одним из основных понятий Теории надежности конструкций является понятие предельного состояния. Условие прочности по существу есть условие обеспечения прочностной надежности.

Основной особенностью реальных услов= ий эксплуатации машин и конструкций является случайный характер взаимодействия= с окружающей средой. Это проявляется в том, что мы не можем достоверно предви= деть все типы внешних нагрузок и их величины, которые могут встретиться в процес= се эксплуатации. Кроме того, источником неопределенности могут быть случайные свойства материалов. Например, предельное напряжение s*, входящее в условие прочности , по своей природе является случайным. Его величина зависит от многих факторов: марки материала, технологии изготовления, размеров детали или конструкции, условий эксплуатации и др. Случайный характер механических свойств материал= ов наглядно проявляется при испытаниях, обнаруживающих значительный разброс экспериментальных данных. Источник неопределенности связан также с разбросом размеров при изготовлении конструкций: в принципе невозможно выдержать абсолютно точно геометрические параметры конструкции, при их изготовлении допускаются некоторые отклонения.

В случае одномерного напряженного состояния

(1)

напряжение s, зависящее от внешних нагрузок, при определенных условиях = может принять довольно большое значение, а предельное значение s* может оказаться малым, так что это неравенство нарушится. Если стечение обстоятельств, приводящее к нарушению условия прочности, редкое событие, то приходим к вероятностной трактовке условия прочности с позиций теории надежности. Вероятностью называет= ся числовая характеристика степени возможности наступления некоторого события в определенных многократно воспроизводимых условиях. Вероятность события А можно оценить на основе опытных данных. Если проводится достаточно большое число опытов N, в которых событие Л= появилось NA раз, то можно считать, что вероятность появления этого события равна

P(A)=3D<= span class=3DGramE>NА/N.

Вероятность как мера возможности наступления события удовлетворяет условиям 0£Р(А)£1= , причем значение Р=3D0 соответствует невозможному событию, а значение= Р=3Dl - достоверному событию.

Вероятность события, заключающегося в выполнении условия (1) Р ( ) в теории надежности называется вероятностью безотказной работы. Вместо условия прочности (1) записывается условие

Р( )=3DР*

(2)

где Р* - заданное достаточно высокое значение вероятности, которое называется нормативной вероятностью безотказной работы. В этом случае говорят, что усл= овие прочности обеспечено с вероятностью Р*= .

Расчетные нагрузки, коэффициенты запаса

Условие прочности (1) записано через напряжения, которые вычисляются через внешние нагрузки, приложенные к конструкции. Пусть внешние нагрузки определены с точностью до одного параметра S, а напряжение = s связано с этим параметром зависимостью

Тогда условие прочности (1) можно записать через внешние нагрузки

S < R

(3)

Здесь через R обозначено предельное значение нагрузки, т.е. такое ее значение, которое приводит к предельному состоянию

Величина R, зависящая от свой= ств материала и условий нагружения, называется несущей способностью или = сопротивлением.

При заданном значении 5 отношение

N =3D R/= S

называется коэффициентом запаса. Он обозначает, что сколько раз нужно увеличить нагрузку, чтобы достичь предельного состояния. Вместо условия прочности (2) можно записать эквивалентное условие

n > 1

(4)

Если нагрузка и свойства материала я= вляются случайными, то условия прочности (3) и (4) теряют смысл, их нужно заменить вероятностными условиями типа (2):

P(S<R= )=3DP*

или

 

P(n >= 1)=3DP*

При этом коэффициент запаса n также будет случайным.

Практически расчет на прочность с уч= етом случайного характера внешних нагрузок и случайных свойств материала проводи= тся следующим образом. Вводится некоторое характерное значение нагрузки [S]<= /i>. Это значение, называемое допускаемым или нормативным значением, можно найти из условия

P(S<[S])=3D[PS]

(5)

где [PS] - некотор= ое значение вероятности, называемое обеспеченностью. Аналогично вводится нормативное значение [R] несущей способности

P(R>[R]=3D[PR]

(6)

Отношение

[n]=3D[R]/[S]

(7)

называется нормативным коэффициен= том запаса. Этот коэффициент зависит от условий нагружения, от свойств материалов, условий работы конструкции, степени ее ответственности и ряда других факторов. Такой коэффициент назначается, исходя из многолетнего опыта эксплуатации конструкций, и для каждого типа конструкций задается нормативно-технической документацией.

В качестве нормативных значений [= S] и [R] можно выбрать средние значения соответствующих случайных велич= ин

где Sj и R= j экспериментально полученные значения случайных величин в серии из N опытов. Однако в действующих нормах, в частности, строительных, нормативные значения не совпадают со средними значениями, а сдвинуты в сторону более опасных значений, что связано со значительным разбросом опытных данных около средних значений. Для нагрузки принимается несколько большее значение, а для несущей способности - меньшее

где коэффициенты cS >1 и cR < 1 находятся из уравнений (5) и (6). Таким образом, нормативный коэффициент запаса (7) вычисляется через средние значения следующим образом:

С учетом случайного характера внешних нагрузок и сопротивлений условие прочности (3) заменяется следующим условием

SP<= /sub> < RP

Здесь SР - достаточно редко встречающееся в реальных условиях эксплуатации высокое значение нагрузки, RР - также достаточно редко встречающе= еся низкое значение несущей способности. Эти значения называются расчетными<= /i>. Они находятся из уравнений

(8)

(9)

В правой части уравнений содержатся нормативные значения вероятности безотказной работы, которые близки к едини= це (0,95; 0,99; 0,999; ...).

Расчетные значения нагрузок и несущей способности можно выразить через средние значения этих величин следующим образом:

где коэффициенты kS >1 и kP < 1 находятся из решения уравнений (8) и (9). Расчетные значения связаны с соответствующими нормативными значениями соотношениями

SP<= /sub> =3D kп[S], RP =3D ko[R]

Коэффициент

kп =3D kS / c= S

называется коэффициентом однородн= ости (меньше единицы). Другой коэффициент, учитывающий случайный характер несущей способности,

kо =3D kR / c= R

называется коэффициентом однородн= ости (меньше единицы).

Это условие можно заменить равенство= м

SP<= /sub>=3DRP/m

где коэффициент m>1 учитыв= ает условия работы конструкции, степень ее ответственности. С учетом обозначени= я (7) для нормативного коэффициента запаса получим формулу, учитывающую случайные свойства нагрузки и несущей способности, а также степень ответственности конструкции

[n] =3D = mkп / kо

Расчеты по допускаемым нагрузкам и по допускаемым напряжениям=

Если пренебречь случайным разбросом прочностных свойств материала конструкции, то расчетное и нормативное значе= ния, а также среднее значение несущей способности R совпадают

RP<= /sub> =3D [R] =3D <R> =3D R

а уравнение (7) позволяет получить выражение нормативной или допускаемой нагрузки через нормативный коэффициент запаса

[S] =3D = R / [n]

При этом параметр несущей способност= и R связан с предельным значением s* напряжения.

Если на заданную конструкцию действу= ет фиксированная неслучайная нагрузка = S, то соотношение

NS<= /sub> =3D R / S

определяет коэффициент запаса по нагрузке. При этом условие прочности можно переписать следующим образом=

S < [= S]

После подстановки условие прочности примет вид

nS<= /sub> > [n]

Переход от нагрузок к вызываемым эти= ми нагрузками напряжениям производится по ранее описанным соотношениям. Отноше= ние

ns =3D s* / s

называется коэффициентом запаса по напряжениям. Можно получить связь между коэффициентами запаса по нагруз= кам и по напряжениям в виде:

В общем случае полученные коэффициен= ты запаса не совпадают, что видно из рис. 1. Равенство этих коэффициентов возможно толь= ко в том случае, когда зависимость между напряжениями и нагрузкой линейна. При нелинейной зависимости коэффициент теряет ясный физический смысл как число,= на которое нужно умножить значение параметра внешней нагрузки, чтобы достичь предельного состояния. По аналогии можно ввести допускаемое напряжение

Расчет по допускаемым напряжениям

в общем случае дает результаты, отли= чные от расчетов по допускаемым нагрузкам. Эти результаты совпадают только в слу= чае линейных зависимостей между напряжениями и нагрузкой.

Следует отметить, что приведенные рассуждения относятся к понятию предельного состояния в точке, которое нужно отличать от предельного состояния конструкции. Предельное состояние в точке= еще не означает потерю несущей способности конструкции. Пусть предельное состоя= ние конструкции будет достигнуто при достижении параметром нагрузки S предельного значения R*. Тогда локальное условие прочности нужно заменить условием

S < R= *

Расчеты с использованием этого услов= ия носят название расчетов по предельному состоянию для конструкции. При этом говорят о конструкционной прочности в отличие от прочности материала, характеризуемой локальным пределом прочности s* или R. Конструкционная прочность зависит не только от прочностных св= ойств материала, но и от масштабного фактора, конструктивной формы, типа напряжен= ного состояния, условий взаимодействия с окружающей средой и ряда других факторо= в.

Растяжение (сжатие) призматических стержней

Ключевые слова: прочность, перемещение, концентрация напряжений, напряженное состояние.

Напряжение при растяжении (сжатии) призматических стержней. Расчет на прочность

Переходя к изучению введенных основн= ых видов деформации стержней, ограничимся рассмотрением стержней постоянного поперечного сечения с прямолинейной осью, т. е. призматических стерж= ней. Начнем с деформации растяжения (сжатия).

Напомним, что под растяжением (сжатием) понимают такой вид деформации стержня, при котором в его поперечном сечении возникает лишь один внутренний силовой фактор - продольн= ая сила Nz. Поскольку продольная сила чи= сленно равна сумме проекций, приложенных к одной из отсеченных частей внешних сил = на ось стержня (для прямолинейного стержня она совпадает в каждом сечении с ос= ью Oz), то растяжение (сжатие) имеет место, если все внешние силы, действующие по о= дну сторону от данного поперечного сечения, сводятся к равнодействующей, направленной вдоль оси стержня (рис. 1). Одна и та же продольная сила Nz при действ= ии на различные части стержня (левую или правую) имеет противоположные направления. Знак Nz зависит от характера вызываемой ею деформации. Продольная сила считается положительной, если вызывает растяжен= ие элемента (рис. 2, а), и она отрицательна, если вызывает сжатие (рис. 2, б).

Для того,= чтобы сформулировать предпосылки теории растяжения (сжатия) призматического стерж= ня, обратимся к эксперименту. Представим себе стержень, изготовленный из какого-либо податливого материала (например, резины), на боковую поверхность которого нанесена система продольных и поперечных рисок (рис. 3, а). Эта ортогональная система рисок остается так= овой и после приложения растягивающей нагрузки (рис. 3, б). Поскольку поперечные риски являются следами поперечных сечений на поверхности стержня и остаются прямыми и перпендикулярными к оси стержня то это свидетельствует о выполнении = гипотезы плоских сечений (Бернулли). С учетом гипотезы об отсутствии поперечн= ого взаимодействия продольных волокон приходим к выводу, что деформация растяжения стержня сводится к одноосному растяжению его продольных волокон,= и в поперечном сечении стержня возникают лишь нормальные напряжения а (рис. 4), индекс г у которых опускаем. Ортогональность продольных и поперечных рисок свидетельствует= также об отсутствии сдвигов, а, следовательно, и связанных с ними касательных напряжений т в поперечных и продольных сечениях стержня.

Тогда продольная сила N" равная сумме проекции внутренних сил, действующих в данном поперечном сечен= ии площадью F (рис. 4) очевидно будет равна

Это соотношение является уравнением равновесия статики, связывающим продольную силу Nz, и нормальное напряжение s, которое в общем случае является функцией координат х и у и поэтому не м= ожет быть найдено из одного лишь 1 уравнения статики . Таким образом, задача определения напряжений даже в самом простом случае деформирования стержня (растяжении или сжатии) оказывается статически неопределимой.

Необходимое для решения этой задачи дополнительное уравнение вытекает из гипотезы плоских сечений. Поскольку поперечные сечения стержня, оставаясь плоскими и перпендикулярными к оси стержня, в процессе деформирования лишь поступательно перемещаются вдоль ос= и стержня (что приводит к одинаковому удлинению всех продольных волокон), то приходим к уравнению e=3Dconst, из которого ввиду однозначности связи s<= /span> и e (для линейно-упругого материа= ла это - закон Гука: s=3DЕe.) вытекает, что

s =3D const.

Решая совместно = уравнения получим, что Nz=3DsF или

s =3D Nz / F.

Таким образом, при растяжении (сжати= и) призматического стержня нормальные напряжения равномерно распределены по поперечному сечению, а касательные напряжения в сечениях отсутствуют, что является следствием гипотезы плоских сечений. Указанное= , несмотря на, казалось бы, очевидность и простоту, является фундаментальным результатом, справедливым, строго говоря, лишь для призматического стержня. Однако в инженерной практике его используют и для приближенной оценки нормальных напряжений в стержнях переменного сечения. При этом, чтобы погрешность формулы была невелика, необходимо,= чтобы площадь поперечного сечения стержня изменялась достаточно плавно вдоль его = оси.

Условие прочности при растяжении (сжатии) призматического стержня для стержня из пластического материала (т.= е. материала, одинаково работающего на растяжение и сжатие) будет иметь вид:

(1)

где [s] - допускаемое напряжение. Напряжение s в условии (1) подставляется по модулю, так как знак s<= /span> в этом случае роли не играет. Для стержней из хрупких материалов, неодинако= во сопротивляющихся растяжению и сжатию, знак напряжения имеет принципиальное значение, и условие прочности приходится формулировать отдельно для растяже= ния и сжатия

где sр и sс - напряжения растяжения и сжатия, а [sр] и [sс] - ответствующие= им допускаемые напряжения.

В практике инженерных расчетов, исхо= дя из условия прочности, решаются три основные задачи механики материалов конструкций. В применении к случаю растяжения (сжатия) призматического стер= жня эти задачи формулируются следующим образом.

Проверка прочности (поверочный расче= т). Этот расчет проводится, если нагрузка (в нашем случае ее представляет N<= sub>z), сечение стержня F и его материал [s] заданы.

Необходимо убедиться, что выполняется условие прочности

Проверочный расчет заключается в том, что определяется фактический коэффициент запаса прочности n и сравнивается с нормативным коэффициентом запаса [n]:

где s* - предельное (или опасное) напряжение, т. е. напряжение, вызывающее отказ элемента конструкции (напомним, что, например, для стержня из пластичного материала это-предел текучести sт или условный п= редел текучести s0,2).

Подбор сечения (проектный расчет). В этом расчете по Заданной нагрузке (Nz) определяются разме= ры поперечного сечения стержня (F) из заданного материала ([s] дано). Минимальное зн= ачение F получим, если в условии прочности (1) принять знак равенства:

[F] =3D = Nz / [s]

Определение допускаемой нагрузки<= /i>, то есть максимального значения нагрузки, которое допускает данный элемент конструкции (F и [s] даны) при выполнении условия прочности (1)

[N] =3D = [s]F

Понятие о концентрации напряжений, принцип Сен-Венана

Даже для призматического стержня равномерное распределение напряжений по поперечному сечению не всегда имеет место. Так, отклонения от равномерного распределения напряжений наблюдаются= в окрестности сечений, содержащих вырезы, выточки, отверстия, трещины, в мест= ах резкого изменения поперечного сечения, а также в местах приложения сосредоточенных сил и т. п. Неравномерное распределение напряжений в указан= ных местах является следствием искажения плоскостей поперечных сечений или их <= i>депланации.

Поясним это явление на примере подверженной растяжению полосы из податливого материала с круговым отверсти= ем, на поверхности которой нанесены продольные и поперечные риски (рис. 5, а). В зоне отверстия имеет место депланация поперечных сечений, вызванная неравномерным растяжением продольных волокон= (рис.5, б). При этом наибольшие удлинения и соответственно напряжения max s получают волокна возле отверстия. Так= ое местное увеличение напряжений возле вырезов, выточек, отверстий и т. п., а также в местах приложения сосредоточенных сил, называется концентрацией напряжений, а источники концентрации напряжений (вырезы, выточки, отвер= стия и т. п.) получили название концентраторов напряжений.

Рассмотренными методами механики деформированного тела, опирающимися на гипотезу плоских сечений, задачи о распределении напряжений в зонах концентрации напряжений не решаются. Такие задачи решаются методами теории упругости или исследуются экспериментально.= При этом для практических расчетов вводится так называемый теоретический коэффициент концентрации напряжений = aк, п= редставляющий собой отношение максимальных max s и номинальных sном напря= жений: aк =3D maxs /s= ном, где номинальные напряжения определяются без учета концентрации напряжений. В приведенном примере растяжения полосы с отверстием sном =3D Nz / Fnt, a= Fnt - площадь поперечного сечения полосы, уменьшенная за счет отверстия ("нетто"). Таким образом, aк играют роль поправочных коэффициентов.

Однако, как показали эксперименты и точные решения задач теории упругости, местные отклонения от равномерного распределения напряжений, вызванные концентрацией напряжений, быстро затуха= ют по мере удаления от сечения с концентратором, и на расстояниях порядка шири= ны сечения распределение напряжений можно считать практически равномерным (рис. 5, в). Отмеченное свойство является частным случаем широко используемого практически во всех разделах механики деформируемого твердого тела (в том числе и теории упругости) принципа Сен-Венана

Определение деформаций и перемещений

Определим упругие деформации стержня предполагая, что изменение его длины при растя= жении Dl, называемое абсолютной продольной деформацией или удлинением, мало по сравнению с его первоначальной длин= ой l(Dl<<l). Тогда относительная продольная деформация будет равна

e =3D Dl / l

Учитывая, что согласно закону Гука д= ля одноосного растяжения (сжатия)

e =3D s / E

где Е - модуль продольной упругости материала стержня, а нормальные напряжения определяются по формул= е -s=3DNz /F (в нашем сл= учае Nz=3DP), для абсолютной деформации получаем

Dl =3D Nz / EF

(2)

Произведение EF принято назыв= ать жесткостью поперечного сечения стержня при растяжении (сжатии), так как удлинение обратно пропорционально EF.

Как показывают эксперименты, при растяжении стержня размеры его поперечного сечения уменьшаются (см. рис. 6), а при сжатии - увеличиваются. Это явление получило название эффе= кта Пуассона.

По аналогии с продольной деформацией изменение размеров поперечного сечения D= b (на рис. 6 Db<0) будем назыв= ать абсолютной поперечной деформацией, а e'= =3DDb/b - относительной поперечн= ой деформацией. Относительные продольная и поперечная = деформа-ции, имеющие противоположные знаки, связаны между собой коэффициентом m, являющимся константой материала и называемым коэффициентом поперечной деформации или коэффициентом Пуассон= а:

e' =3D - me

Как известно, для изотропного матери= ала 0 £ m £1/2.

Формула (2) для удлинения стержня Dl применима только в случае, когда по длине стержня ни жесткость поперечного сечения, ни продольная сила не изменяются (EF=3Dconst, Nz=3Dconst). Удлинение стержня со ступенчатым изменением EF и Nz (рис. 7) может быть определено как сумма удлинений ступеней, у которых EF<= /i> и Nz постоянны:

(индекс k у модуля продольной упругости означает, что участки стержня могут быть изготовлены из различных материалов). В случае, когда Nz и EF меняются по д= лине стержня l непрерывно и их можно считать постоянными лишь в пределах ступеней длиной dz, обобщая формулу эту, получаем

В качестве тестов для практики расчетов определенных интегралов рекомендую воспользоваться системой входных тестов Т-5, указанных в ПРИЛОЖЕНИИ.

С упругими продольными деформациями стержня при растяжении (сжатии) связаны продольные перемещения его сечений.= На рис. 8 приведены три случая определения таких перемещений, откуда видно, что перемещения поперечных сечений численно равны удлинениям заштрихованных час= тей стержня:

  • пере= мещение свободного торцевого сечения 1-1 при неподвижном другом торцевом сечен= ии (рис. 8, а) численно равно удлинению стержня;
  • перемещение промежуточного сечения 2= -2 (рис. 8, б) численно равно удлинению части стержня, заключенной между да= нным сечением и сечением неподвижным;
  • взаи= мное перемещение сечений 3-3 и 4-4 (рис, 8, в) численно равно удлинению части стержня, заключенной между этими сечениями.

Напряженное состояние при растяжении (сжатии)

Напряженное состояние при растяжении стержня является одноосным (рис. 9, а). Поскольку на поперечных и продольных площадках касательные напряжения не возникают, то эти площадки являются главными. Причем в случае растяжения s1 =3Ds >= ;0, s2=3Ds3=3D0, а в = случае сжатия s1=3Ds2=3D0, а s3=3Ds<0.

Напряжения на площадках, наклоненных= к оси стержня под углом a, определя= ются по формулам для упрощенного плоского напряженного состояния:

Площадки с экстремальными касательны= ми напряжениями t13 (рис. 9, б), как известно, наклонены по отношению к исход= ным под углами b=3D±45° (следует и из= формулы для ta) и равны t13=3Ds/2.

Именно с действи= ем экстремальных t связывается появл= ение на боковой поверхности образца из малоуглеродистой стали, испытываемого на растяжение, линий скольжения, ориентированных под углом b=3D±45° к оси образца. На площадках с экстремальными t действуют и нормальные напряжения, равные s13=3Ds/2.

Прямой чистый изгиб призматического стержня

Ключевые слова: прочность, жесткость, осевой момент инерции, осевой момент сопротивления.

При прямом чистом изгибе в поперечном сечении стержня возникает только один силовой фактор - изгибающий момент Мх (рис. 1). Так как Qy=3DdMx/dz=3D0, то Mx= =3Dconst и чистый прямой изгиб может быть реализован при загружении стержня парами с= ил, приложенными в торцевых сечениях стержня. Поскольку изгибающий момент M<= sub>х по определению равен сумме моментов внутренних сил относительно оси Ох с нормальными напряжениями его связы= вает выкающее из этого определения уравнение статики

Сформулируем предпосылки теории чист= ого прямого изгиба призматического стержня. Для этого проанализируем деформации модели стержня из низкомодульного материала, на боковой поверхности которого нанесена сетка продольных и поперечных рисок (рис. 2). Поскольку поперечные риски при изгибе стержня парами сил, приложенн= ыми в торцевых сечениях, остаются прямыми и перпендикулярными к искривленным продольным рискам, это позволяет сделать вывод о выполнении гипотезы пло= ских сечений, которая, как показывает решение этой задачи методами теории упругости, перестает быть гипотезой, становясь точным фактом - законом плоских сечений. Замеряя изменение расстояний между продольными рисками, приходим к выводу о справедливости гипотезы о ненадавливании продольных вол= окон (sх =3Dsy=3D0).

Ортогональность продольных и попереч= ных рисок до и после деформирования (как отражение действия закона плоских сече= ний) указывает также на отсутствие сдвигов, касательных напряжений в поперечных и продольных сечениях стержня.

Таким образом, чистый прямой изгиб призматического стержня сводится к одноосному растяжению или сжатию продоль= ных волокон напряжениями s (индекс г в дальнейшем опускаем). При этом часть волокон находится в зоне растяжения (на рис. 2 это - нижние волокна), а другая часть - в зоне сжатия (верхние волокн= а). Эти зоны разделены нейтральным слоем (n-n), не меняющим своей длины, напряж= ения в котором равны нулю. Учитывая сформулированные выше предпосылки и полагая,= что материал стержня линейно-упругий, т. е. закон Гука в этом случае имеет вид:= s=3DeЕ, выведем формулы для крив= изны нейтрального слоя 1/r (r - радиус кривизны) и нормальных напряжений s. Предварительно отметим, что постоянство поперечного сечения призматического стержня и изгибающего момента (Mх= =3Dсonst), обеспечивает постоянство радиуса кривизны нейтрального слоя по длине стержн= я (рис. 3, а), нейтральный слой (n-n) описывается дугой окружности.

Рассмотрим призматический стержень в условиях прямого чистого изгиба (рис. 3, а) с поперечным сечением, симметричным относительно вертикальной оси= Оу. Это условие не отразится на конечном результате (чтобы прямой изгиб был возможен, необходимо совпадение оси Оу с главной осью инерции поперечного сечения, которая и является осью симметрии). Ось Ox поме= стим на нейтральном слое, положение которого заранее неизвестно.

Рассмотрим вырезанный из стержня эле= мент длиной dz, который в масштабе с искаженными в интересах наглядности пропорциями изображен на рис. 3, б. Поскольку интерес представляют деформации элемента, определяемые относительным смещением его точек, одно из торцевых сечений элемента можно считать неподвижным. Ввиду малости dj считаем, что точки поперечного сечения при повороте на этот угол перемещают= ся не по дугам, а по соответствующим касательным.

Вычислим относительную деформацию продольного волокна АВ, отстоящего от нейтрального слоя на у:

e =3DВВ1 / АВ=3DВВ1 / ОО1

Из подобия треугольников С00= 1 и 01ВВ1 следует, что

BB1= / OO1 =3D O1B/CO =3D y /r

Продольная деформация e оказалась линейной функцией расстояни= я от нейтрального слоя, что является прямым следствием закона плоских сечений

(1)

Тогда нормальное напряжение, растягивающее волокно АВ, на основании закона Гука будет равно

(2)

Эта формула не пригодна для практического использования, так как содержит две неизвестные: кривизну нейтрального слоя 1/r и положение нейтральной оси Ох, от к= оторой отсчитывается координата у. Для определения этих неизвестных воспользуемся уравнениями равновесия статики. Первое выражает требование равенства нулю продольной силы

(3)

Подставляя в это уравнение выражение= (2)

и учитывая, что = (Е / r) ¹ 0, получаем, что

Интеграл в левой части этого уравнен= ия представляет собой статический момент поперечного сечения стержня относител= ьно нейтральной оси Ох, который может = быть равным нулю только относительно центральной оси. Поэтому нейтральная ось Ох проходит через центр тяжести попереч= ного сечения.

Вторым уравнением равновесия статики является, связывающее нормальные напряжения с изгибающим моментом (который легко может быть выражен через внешние силы и поэтому считается заданной величиной). Подставляя в уравнение связки выраже= ние для. напряжений, полу= чим:

и учитывая, что где Jx - главный центральный момент инерции относительно оси Ох, для кривизны нейтрального слоя получаем формулу

(4)

Кривизна нейтрального слоя 1/s является мерой деформации стержня при прямом чистом изгибе. 1/s тем мен= ьше, чем больше величина EJх, назыв= аемая жесткостью поперечного сечения при изгибе (по аналогии с жесткостью попереч= ного сечения при растяжении EF).

Подставляя (4) в (2), получаем формулу для нормальных напряжений в виде

(5)

которая была впервые получена Ш. Кул= оном в 1773 году. Для согласования знаков изгибающего момента Мх и нормальных напряжений s в правой = части формулы (5) ставится знак минус, так как при Mх>0 нормальные напряжения s при y&= gt;0 оказываются сжимающими. Однако в практических расчетах удобнее, не придерживаясь формального правила знаков, определять напряжения по модулю, а знак ставить по смыслу. Нормальные напряжения при чистом изгибе призматичес= кого стержня являются линейной функцией координаты у и достигают наибольших знач= ений в волокнах, наиболее удаленных от нейтральной оси (рис. 4), т. е.

Здесь введена геометрическая характеристика , имеющая размерность м3 и получившая название моме= нта сопротивления при изгибе. Поскольку при заданном Mх напряжения max s тем меньш= е, чем больше Wx, момент сопротивления является геометрической характеристикой прочности поперечного сечения изгибе. Приведем примеры вычисления моментов сопротивления для простейших форм поперечных сечений. Д= ля прямоугольного поперечного сечения (рис. 5, а) имеем Jх=3Dbh3/12, ymax =3D h/2 и Wx =3D Jx/ymax<= /sub> =3D bh2/6. Аналогично для круга (рис. 5,6 Jx=3Dp<= i>d4/64, ymax=3Dd/2) получаем Wx=3Dpd3/32, для кругового кольцевого сечения (= рис. 5, в), у которого

получаем

Итак, максимальн= ые нормальные напряжения в сечении с изгибающим моментом Mх определяются по формуле

(6)

Этой формулой удобно пользоваться для расчета балок пластичного материала в упругой области, одинаково работающег= о на растяжение и сжатие. Поскольку знак напряжения в этом случае не имеет значе= ния, напряжения вычисляются по модулю, и условие прочности при изгибе балки в фо= рме призматического стержня получает вид

где max M= х - максимальное значение изгибающего момента (легко определяемое по его эпюр= е), [s] - допускаемое напряжение на п= ростое растяжение (сжатие). Напомним, что чистый изгиб балки сводится к растяжению= и сжатию ее волокон (неравномерному в отличие от деформации растяжения (сжати= я) призматического стержня, при котором s=3Dconst).

При расчете балок из хрупких материа= лов следует различать наибольшие растягивающие max sp и наибольшие сжимающие max |sc| напряжения (рис. 6), которые также определяются по модулю непосредственно и сравниваются= с допускаемыми напряжениями на растяжение |sт| и сжатие [sс]. Условие прочности в этом случае будет иметь вид:

Прямой поперечный изгиб призматического стержня

Ключевые слова: прочность, жесткость, двутавр, швеллер, рациональное сечение, равнопрочность.

При прямом поперечном изгибе в сечен= иях стержня возникает изгибающий момент Мх и поперечная сила Qy (рис. 1), которые связаны с нормальными s и касательными tyz напряжениями

Выведенная в случае чистого изгиба стержня формула для прямого поперечного изгиба, вообще говоря, неприменима, поскольку из-за сдвигов, вызываемых касательными напряжениями tyz, происходит депланация поперечных сечении (отклонение от закона плоских сечений). Однако для балок= с высотой сечения h<l/4 (рис. 2) погрешность невелика и ее применяют для определения нормальных напряжений поперечного изгиба как приближенную. При выводе условия прочности при чистом изгибе использовалась гипотеза об отсутствии поперечного взаимодействия продольных волокон. При поперечном изгибе наблюдаются отклон= ения от этой гипотезы:

а) в местах приложения сосредоточенн= ых сил. Под сосредоточенной силой напряжения поперечного взаимодействия могут = быть достаточно велики и во много раз превышать продо= льные напряжения sz, убывая = при этом, в соответствии с принципом Сен-Венана, по мере удаления от точки приложения силы;

б) в местах приложения распределенных нагрузок. Так, в случае, приведенном на рис. 2, б, напряжения от давления на верхние волокна балки sy =3D -q/b. Сравнивая их с прод= ольными напряжениями sz, имеющ= ими порядок

приходим к выводу, что напряжения sy << sz при условии, что h2 << l2, так как sy /sz (h/l)2 << 1.

Получим формулу для касательных напряжений tyz. Примем, методика расчета нормальных напряжений известна, что касательные напряжения равномерно распределены по ширине поперечного сечения (рис. 3). Эта предпосылка выполняется тем точнее, чем уже поперечное сечение стержня. Точное решение задачи для прямоугольного поперечного сечения показывает, что отклонение от равномерного распределения tyz , зависит от отношения сторон b/h. При (b/h)=3D1,0 оно составляет 12,6%, п= ри (b/h)=3D0,5 - только 3,3%.

Непосредственное определение напряже= ний tyz затруднительно, поэтому находим равные им (вследствие закона парности) касательные напряжения tyz, возникающие на продольн= ой площадке с координатой у элемента длиной dz, вырезанного из балки, (= рис. 3). Сам элемент показан на рис. 4. От этого элемента продольным сечением, отстоящим от нейтрального сло= я на у, отсекаем верхнюю часть, заменяя действие отброшенной нижней части касательными напряжениями t (инде= кс гу в дальнейшем опускаем), равнодействующая которых dT=3Dtbdz показана на рис. 5. Здесь, согласно второй предпосылке

t= =3Dconst по ширине элемента b. Нормальные напряжения s и s + ds, действующие на торцевых площадках элемента, также заменим их равнодействующими

Согласно первой предпосылке нормальные напряжения определяются уже известным способом, , где Sxw - статический момент отсеченной части площади поперечного сечения w относительно оси = Ох.

Рассмотрим условие равновесия элемен= та (рис. 5) составив для него уравнение статики Sz =3D 0:

откуда после несложных преобразовани= й, учитывая, что

получаем формулу для касательных напряжений при нормальном поперечном изгибе призматического стержня

которая называется формулой Журавского. В этой формуле by - ширина сечения в том месте, где определяются касательные напряжения, а статический момент, подставляемый в эту формулу, может быть вычислен как для верхней, т= ак и для нижней части (статические моменты этих частей сечения относительно его центральной оси Ох отличаются толь= ко знаком, так как статическим момент всего сечения равен нулю).

В качестве примера применения формулы Журавского построим эпюру касательных напряжений для случая прямоугольного поперечного сечения балки (рис. 6). Учитывая, что для этого сечения

получаем

где F=3Dbh - площадь прямоугольника.

Как видно из формулы, касательные напряжения по высоте сечения меняются по закону квадратичеокой параболы, достигая максимума на нейтральной оси

Сделаем несколько замечаний, касающи= хся расчетов на прочность при прямом поперечном изгибе. В отличие от простых ви= дов деформации, когда в поперечных сечениях стержня возникает лишь один силовой фактор, к которым относятся и изученные выше растяжение (сжатие) и чистый изгиб, прямой поперечный изгиб должен быть отнесен к сложным видам деформац= ии. В поперечных сечениях стержня при поперечном изгибе возникают два силовых фактора: изгибающий момент Мх и поперечная сила Qy (рис. 7), напряженное состояние является упрощенным плоским, при котором в окрестности произвольно выбранных точек поперечного сечения действуют нормальные s и касательные t напряжения. Поэтому условие прочности для таких точек должно быть сформулировано на основе какого-либо уже известного крите= рия прочности.

Однако учитывая, что наибольшие нормальные напряжения возникают в крайних волокнах, где касательные напряжения отсутствуют (рис. 7), а наибольшие касательные напряжения во многих случаях имеют место в нейтральном слое, где нормальные напряжения равны нулю, условия прочности в этих случаях формулируются раздельно по нормальным и касательным напряжения= м

Покажем, что доминирующая роль в расчетах на прочность балки, подвергнутой поперечному изгибу, будет принадлежать расчету по нормальным напряжениям. Для этого оценим порядок max s и max t на примере консольной балки, показанной на рис. 8:

так как Тогда

откуда max t<<maxs, а поскольку [t]/[s] 0,5 то доминирующим в этом случае будет расчет по нормальным напряжениям и условие прочности, например, для балки из пластичного материа= ла, работающей на прямой изгиб, как и в случае чистого изги= ба будет иметь вид

Рациональные формы поперечных сечений при изгибе

Наиболее рациональным следует призна= ть сечение, обладающее минимальной площадью при заданной нагрузке (изгибающем моменте) на балку. В этом случае расход материала на изготовление балки, бу= дет минимальным. Для получения балки минимальной материалое= мкости нужно стремиться к тому, чтобы по возможности наибольший объем материала работал при напряжениях, равных допускаемым или близким к ним. Прежд= е всего рациональное сечение балки при изгибе должно удовлетворять условию равнопрочности растянутой и сжатой зон балки. Иными словами необходимо, чтобы наибольшие напряжения растяжения (max sр) н наибольшие напряжения = сжатия (max sс) одновр= еменно достигали допускаемых напряжений [sр] и [s<= /span>с].

Поэтому для балки из пластичного материала (одинаково работающего на растяжение и сжатие: [sр] =3D [sс] =3D [s]), условие равнопрочности выполняется для сечений, симметричных относительно нейтральной оси. К таким сечениям относится, например, прямоугольное сечение (рис. 9, а), при котором обеспечено условие равенства maxsр=3Dmaxsс. Однако в этом случае мат= ериал, равномерно распределенный по высоте сечения, плохо используется в зоне нейтральной оси. Чтобы получить более рациональное сечение, необходимо возм= ожно большую часть материала переместить в зоны, максимально удаленные от нейтральной оси. Таким образом, приходим к рациональному для пластичного материала сечению в форме симметричного двутавра (рис. 9, б), у которого возможно большая часть материала сосредоточена на пол= ках (горизонтальных массивных листах), соединенных с= тенкой (вертикальным листом), толщина которой (= d) назначается из условий прочности стенки по касательным напряжениям, а также= из соображений ее устойчивости (см. гл. 15). К двутаврому сечению близко по критерию рациональности так называемое коробчатое сечение (рис. 9, в).

Рассуждая аналогично, приходим к выв= оду, что для балок из хрупкого материала наиболее рациональным будет сечение в ф= орме несимметричного двутавра, удовлетворяющего условию равнопрочности на растяж= ение и сжатие (рис. 10):

которое вытекает из требования

Идея рациональности поперечного сече= ния стержней при изгибе реализована в стандартных тонкостенных профилях, получа= емых методами горячего прессования или прокатки из рядовых и легированных конструкционных высококачественных сталей, а также алюминия и алюминиевых сплавов, получивших широкое распространение в строительстве, машиностроении, авиационном машиностроении. Широко распространены показанные на рис. 11: а - двутавр, б - швеллер, в - неравнобокий уго= лок, г-равнобокий уголок. Реже встречаются тавр, таврошвеллер, зетовый профиль и= др. Употребляются также холодногнутые замкнутые сварные профили.

Поскольку по соображениям технологии сортамент стандартных профилей по размерам ограничен (например, наибольший прокатный двутавр согласно ГОСТ 8239-72 имеет высоту 550 мм), то для больших пролетов приходится применять составные (сварные или клепаные) балки.

Составные балки и перемещения при изгибе

Ключевые слова: сварные двутавровые балки, уравнение упругой кривой, прогиб, угол поворота, граничн= ые условия.

Понятие о составных балках

Работу составных балок проиллюстриру= ем на простом примере трехслойной балки прямоугольного поперечного сечения. Ес= ли слои между собой не связаны и силы трения между ними отсутствуют, то каждый= из них деформируется как отдельная балка, имеющая свой нейтральный слой (рис. 1, а). Нагрузка между этими балками распределяется пропорционально их жесткостям при изгибе (в данном примере поровну). Это означает, что моменты инерции и моменты сопротивления трех независимо друг от друга деформирующих= ся балок должны быть просуммированы

Если скрепить балки сваркой, болтами= или другим способом (рис. 1, б), то с точностью до пренебрежения податливостью наложенных связей сечение балки будет работать как монолитное с моментом инерции и моментом сопротивления, равным

Как видно, при переходе к монолитному сечению жесткость балки возрастает в девять раз, а прочность - в три раза. В инженерной практике наиболее распространены сварные двутавровые балки.

Дифференциальное уравнение прямого изгиба призматического стержня

Определено, что мерой деформации призматического стержня при прямом чистом изгибе является кривизна нейтраль= ного слоя. Можно показать, что с достаточной для инженерных расчетов точностью э= тим тезисом можно пользоваться и в случае прямого поперечного изгиба стержня. Однако для практических целей кроме кривизны 1/r необходимо определить вертикальные перемещения центров тяжести отдельных поперечных сечений - прогибов балки v, а иногда и углы поворота этих сечени= й j (рис. 2). Вследствие гипотезы плоских сечений угол поворо= та сечения (j оказывается равным углу наклона касательной к изогнутой оси балки, который в силу малости

Тогда возникает геометрическая задач= а: составить уравнение для функции прогиба , зная закон изменения ее кривизны.

Воспользуемся известным из дифференциальной геометрии выражением для кривизны в прямоугольных декартов= ых координатах:

Однако, учитывая, что в инженерной практике применяются достаточно жесткие балки, для которых наибольший проги= б f мал по сравнению с длиной (f / l << 1), а первая производная от прогиба имеет порядок

и, следовательно, величиной (dv /= dz)2<<1, стоящей в знаменателе, можно пренебречь, выражение для кривизны упрощается<= /p>

Тогда, подставив это выражение в полученную ранее связку кривизны и изгибающего мометна - , условившись что ось Oy направ= лена вверх и согласовав знаки 1/r и Мх, приходим к дифференциальному уравнению прямого изгиба балки

известному также как дифференциальное уравнение упругой кривой.

Если учесть точное выражение для кривизны по формуле, то точное уравнение упругой кривой

является нелинейным дифференциальным уравнением. Поэтому линейное дифференциальное уравнение, описывающее малые прогибы балки, иногда называют линеаризованным уравнением упругой кривой= .

Решение уравнения получаем путем двукратного почленного интегрирования. При первом интегрировании получаем выражение

которое с учетом , дает также закон изменения углов поворота поперечных сечений по длине балки. Повторным интегрированием получаем функцию прогиба

Постоянные интегрирования С и D должны быть найдены из граничных условий.

Во всех приведенных выше уравнениях функция изгибающего момента Мх(г)<= /i> предполагалась известной, что возможно лишь для статически определимых бало= к. Простейшие варианты статически определимых однопролетных балок и соответствующие граничные условия показаны на рис. 3. Условия, накладываемые на прогиб и угол поворота сечения, получили название кинематических граничных условий. Как видно, для шарнирно опертой балки требуется, чтобы прогиб на опорах v(0) =3Dv(l) =3D0, а= для консольной балки прогиб и угол поворота сечения в заделке

Дифференциальное уравнение непримени= мо для расчета статически неопределимых балок, так как содержит неизвестный изгибающий момент Мx появивший= ся в результате двукратного интегрирования уравнения четвертого порядка

В этом уравнении нагрузка q известна, поэтому его можно получить, учитывая, что

При интегрировании уравнения необход= имо задать четыре граничных условия (по два на каждом конце балки) в том числе = так называемые силовые граничные условия - условия, накладываемые на силовые величины (изгибающий момент и поперечную силу), которые выражаются через производные от прогиба. Так как

а с учетом дифференциального соотнош= ения Qy=3DdMx/dz, получаем

Вернемся к интегрированию уравнения = второго порядка. Если имеется несколько участков, для которых правая часть уравнения исходного f(z)=3DMx/EJx, содержит разные аналитические выражения, то интегрирование усложняется. На рис. 4 приведена эпюра Мx, соде= ржащая n участков. Для каждого участка независимое интегрирование дает по д= ве константы, а при п участках требуется определить= 2n постоянных. Добавляя к двум граничным условиям на опорах 2(n-1) усло= вия непрерывности и гладкости упругой кривой на границе; смежных участков, заключающиеся в равенстве прогибов v и углов поворота сечений dv/= dz на этих границах

получим 2n граничных условий, необходимых для нахождения постоянных интегрирования.

Рекомендую для практики решения дифференциальных уравнений второго порядка воспользоваться системой входных тестов Т-6, приведенных в ПРИЛОЖЕНИИ.

Напряжения и деформации при кручении призматических стержней кругового поперечного сечения

Ключевые слова: чистый сдв= иг, жесткость сечения при кручении, угол закручивания, вал, прочность, жесткост= ь.

Кручением называется такой вид деформации, при котором в поперечном сечении стержня возникает лишь один силовой фактор - крутящий момент Мz. Крутящий момент по определению равен сумме моментов внутренних сил относите= льно продольной оси стержня Oz. Нормальные силы, параллельные оси Oz, вклада в крутящий момент не вносят. С силами, лежащими в плоскости поперечн= ого сечения стержня (интенсивности этих сил - касательные напряжения txz и tyz) Мz<= /i> связывает вытекающее из его определения уравнение равновесия статики (рис. 1)

Условимся считать Mz положительным, если со стороны отброшенной части стержня видим его направле= нным против часовой стрелки (см. рис. 2). Это правило проиллюстрировано на рис. 1 и в указанном соотношении, где крутящий момент Мz принят положительным. Численно крутящий момент равен сумме моментов внешних сил, приложенных к отсеченной части стержня, относительно оси Оz.

Рассмотрим кручение призматических стержней кругового поперечного сечения. Исследование деформаций упругого стержня с нанесенной на его поверхности ортогональной сеткой рисок (рис. 3) позволяет сформулировать следующие предпосылки теории кручения этого стержня:

  1. попе= речные сечения остаются плоскими (выполняется гипотеза Бернулли);
  2. расс= тояния между поперечными сечениями не изменяются, следова= тельно ez=3D0;
  3. конт= уры поперечных сечений и их радиусы не деформируются. Это означает, что поперечные сечения ведут себя как жесткие круговые пластинки, поворачивающиеся при деформировании относительно оси стержня Оz. Отсюда следует, что любые деформации в пло= скости пластинки равны нулю, в том числе и ex =3D ey =3D0;
  4. мате= риал стержня подчиняется закону Гука. Учитывая, что ex =3Dey =3D ez =3D0, из о= бобщенного закона Гука в форме получаем ex =3Dsy =3D sz =3D0. Это означает, = что в поперечных сечениях, стержня возникают лишь касательные напряжения<= /i> t, а вследствие закона парности касательных напряжений, равные им напряжения действуют и в сопряженных продольных сечениях. Следовательно напряжен= ное состояние стержня - чистый сдвиг.

Выведем формулу для касательных напряжений при кручении призматического стержня кругового поперечного сечен= ия. Как видно, поворот правого торцевого сечения относительно неподвижного лево= го на угол j (назовем его углом закручивания стержня) вызывает поворот продоль= ных волокон на угол g (угол сдвига), поскольку на величину g искажаютс= я углы ортогональной сетки продольных и поперечных рисок модели.

Двумя смежными сечениями вырежем эле= мент стержня длиной dz и, поскольку нас интересуют деформации элемента, л= евое сечение его будем считать неподвижным (рис. 4). При повороте правого сечения на угол dj в соответствии с гипотезой о недеформируемости радиусов, правый конец волок= на АВ (отстоящий от оси элемента на величину полярного радиуса r) будет перемещаться по дуге BB1, вызывая поворот волокна на угол сдвига

Обратим внимание на то, что в соответствии с рис. 4 и рис. 5, а сдвиг g и связанное с ним касательное напряжение t перпендикулярны радиусу r. Определим t, воспользовавшись зак= оном Гука для чистого сдвига

(1)

Здесь dj /dz - погонный угол закручивания стержня, который остается пока неизвестным. Для его нахождения обратимся к условию статики, записав его в более удобной для данного случая форме (рис. 5, a)

(2)

Подставляя (1) в (2) и учитывая, что

где Jp - полярный момент инерции поперечного сечения (для круга с диаметром d Jp=3Dp4/32), получаем

(3)

Подставляя выражение (3) в (1), получаем формулу для касательных напряжений при кручении призматического стержня кругового поперечного сечения

(4)

Как видно из (4), сдвиги и касательные напряжения пропорциональны расстояний от оси стержня. Обратим внимание на структурные аналогии формул для нормальных напряжений чистого изгиба и касательных напряжений кручения.

Мерой деформации стержня при кручении является погонный угол закручивания стержня, определяемый по (3). Поскольку величина DJp стоит в знаменателе формулы и при заданной нагрузке (Mz через нее выражается) dj /dz тем меньше, чем больше = DJp, последнюю называют жесткостью поперечного сечения при кручении.

Пользуясь (3) для определения угла закручивания элемента длиной dz

найдем полный угол закручивания стер= жня длиной l

(5)

В случае,= если по длине стержня Мz и DJp постоянны, получаем

когда эти величины кусочно-постоянны, то:

(6)

Отметим, что полученные формулы по структуре аналогичны формулам для деформаций при растяжении стержня.

Наибольшие касательные напряжения возникают у внешней поверхности стержня, т. е. при rmax=3Dd/2

где Wр - момент сопротивления при кручении или полярный момент сопротивления

Полярный момент сопротивления, стоящ= ий в знаменателе для максимальных касательных напряжений, очевидно, является геометрической характеристикой сечения, а условие прочности стержня при кручении принимает вид

(7)

где [t] - допускаемое напряжение на кручение.

Как показали эксперименты и точное решение этой задачи в теории упругости, все гипотезы, сформулированные ранее для стержня со сплошным круговым сечением, остаются справедливыми и для сте= ржня кольцевого поперечного сечения (рис. 6). Поэтому все выведенные ранее формулы пригодны для расчета стержня кольцевого сечения с той лишь разницей, что полярный момент инерции определяется как разность моментов инерции кругов с диаметрами D и <= i>d

где b=3Dd/D, а момент сопротивления определяется по формуле

Учитывая линейный характер изменения касательных напряжений по радиусу (рис. 6) и связанное с этим лучшее использование материала, кольцевое сечение следует признать наиболее рациональным при кручении стержня. Коэффициент использования материала тем выше, чем меньше относительная толщина трубы.

Как отмечено ранее, напряженное состояние при кручении стержня - чистый сдвиг, являющийся частным случаем плоского напряженного состояния. На площадках, совпадающих с плоскостью поперечного сечения и на парных им площадках продольных сечений возникают экстремальные касательные напряжения max-min t, а главные напряжения s= 1,3 =3D ± t действуют на площадках, наклоненных .коси стержня под углами ±45°; главн= ое напряжение s2 =3D 0.

Особенности напряженного состояния п= ри кручении нашли отражение в характере разрушения стержней. Так, разрушение стержня из дерева, плохо работающего на скалывание вдоль волокон, происходи= т от продольных трещин (рис. 7, a). Разрушение стержня из хрупкого металла (например, чугуна) происх= одит по винтовой линии, наклоненной к образующим под углом 45°, т. е. по траекто= рии главного напряжения s3= (рис. 7,б).

Расчет валов

Рассмотрим расчет вала на прочность и жесткость. Пусть известна мощность W (кВт), передаваемая вращающимся= с заданным числом оборотов в минуту (n) валом от источника мощности (например, двигателя) к ее потребителю (например, станку), а момент m, передаваемый валом, требуется найти, так как численно равный этому моменту крутящий момент необходим для расчета вала.

Если число оборотов вала в минуту п и соответствующая угловая скорость w(с-1) постоянны, а Ф - угол поворота вала в данный момент времени t= , то работа вращательного движения А=3DmФ. Тогда передаваемая валом мощность будет равна

Отсюда

кНм

где учтено, что .

Если мощность подается на вал через ведущий шкив, а раздается потребителям через нес= колько ведомых шкивов, то соответственно определяются м= оменты на шкивах, а затем строится эпюра крутящих моментов. Расчет вала на прочнос= ть и жесткость ведется, очевидно, по max Mz.

Определение диаметра вала из усло= вия прочности. Условие прочности при кручении вала имее= т вид (7), где допускаемые напряжения [t] принимаются пониженными по сравнению с допускаемыми напряжениями обычного статического расчета в связи с необходимостью учета наличия концентраторов напряжений (например, шпоночных канавок), переменного характера нагрузки и наличия наряду с кручением и изг= иба вала.

Требуемое значение Wp= =3Ddз/16 получаем из условия (7), принимая в нем знак равенства

откуда получаем формулу для диаметра вала кругового сечения

Определение диаметра вала из усло= вия жесткости. Условие жесткости состоит в наложении ограничения на погонный угол закручивания вала , так как недостаточно жесткие валы не обеспечивают устойчивой передачи мощности и подвержены сильным колебаниям:

Тогда, учитывая, что Jp=3Dpd4/32, для диаметра = вала из условия жесткости имеем

Аналогично проводятся расчеты и для = вала кольцевого поперечного сечения.

Сложные виды деформации

Ключевые слова: косой изги= б, внутреннее сжатие-растяжение, условия прочности.

Принцип независимости действия сил и границы его применения

Вид деформации является сложным, ког= да в поперечном сечении стержня возникают два и более силовых факторов. Сложный = вид деформации можно рассматривать как сумму простых видов, изученных ранее (растяжение, изгиб, кручение), если применим принцип независимости действия= сил (частный случай принципа суперпозиции или наложения, применяемый в механике деформируемого твердого тела).

Напомним формулировку принципа независимости действия сил: напряжение (деформация) от группы сил равно сумме напряжений (деформаций) от каждой силы в отдельности. Он справедл= ив, если функция и аргумент связаны линейной зависим= остью. В задачах механики материалов и конструкций становится неприменимым, если:<= /p>

  • напр= яжения в какой-либо части конструкции от одной из сил или группы сил превышают предел пропорциональности s<= sub>пц;
  • дефо= рмации или перемещения становятся настолько большими, что нарушается линейная зависимость между ними и нагрузкой.

Например, дифференциальное уравнение изгиба стержня является нелинейным и вытекающая из него зависимость прогиба= f от нагрузки Р для консольной балки, изображенной на рис. 1, а, также является нелинейной (рис. 1, б). Однако, если прогибы балки невелики (f<<l) настольк= о, что (dv/dz)2<<1 (так как dv/dz ~ f/l), то дифференциальное уравнение изгиба становится линейным (как видно из рис. 1, б, начальный участок зависимости Р= от f, описываемый этим уравнением, также является линейным).

Косой изгиб призматического стержня

Известно, что косой изгиб имеет мест= о, когда силы, его вызывающие, не лежат в одной из главных плоскостей инерции. Однако, если разложить внешние силы по главным осям инерции Ох и Оу, то получим две системы сил P1x, P2x, = ... , Pnx и P1y, P2y, ... , Pny, каждая из которых вызывает прямой изгиб с изгибающими моментами соответстве= нно My и Мx (рис. 2). Применяя принцип независимости действия сил, нормальные напряжения = s (рис. 3) определим как алгебраическую сумму напряжений от Mx и Мy:

Чтобы не связывать себя формальными правилами знаков, слагаемые будем определять по модулю, а знаки ставить по смыслу. Прогибы балки определим как геометрическую сумму прогибов от прямых изгибов (рис. 2)

Таким образом, расчет на косой изгиб= с применением принципа независимости действия сил сводится к расчету на два прямых изгиба с последующим алгебраическим суммированием напряжений и геометрическим суммированием прогибов.

 

В случае поперечных сечений, имеющих= две оси симметрии и выступающие угловые точки (рис. 4) с равными по модулю и максимальными одноименными координатами и напряжения в этих точках будут равны

Слагаемые в этом выражении рекоменду= ется определять по модулю, а знаки ставить по смыслу. Например, на рис. 5 верхний ряд знаков "+" и "-= " соответствует напряжениям от Мx, а нижний ряд - от My, и напряжения в этих точках будут равны

Условие прочности для балок из пластичного материала с указанным типом сечений запишется в виде

В остальных случаях для определения = max а (или max dp и max | = sc | для хрупкого материала) необходимо по общей формуле проверить напряжения = во всех подозрительных точках.

Есть и другой путь: положив s =3D 0, получим уравнение нейтральной = линии. Так как напряжения в точках поперечного сечения будут пропорциональными расстояниям от нейтральной линии, то max s будут возникать в наиболее удаленных от нее точках.

Сочетание изгиба и кручения призматического стержня

Ключевые слова: вал, эквивалентный момент, эквивалдентные напряжения, прочность.

Исследуем этот вид деформации стержн= я на примере расчета вала кругового (кольцевого) поперечного сечения на совместн= ое действие изгиба и кручения (рис. 1).

Примем следующий порядок расчета.

1. Разлагаем все внешние силы на составляющие

P1x= , P2x, ... , Pnx и P1y, P2y, ... , Pny=

2. Строим эпюры изгибающих момент= ов My и My. от этих групп сил.

У кругового и кольцевого поперечного сечений все центральные оси главные, поэтому косого изгиба у вала вообще не может быть, следовательно, нет смысла в каждом сечении иметь два изгибающих момента Mx, и My а целесообразно их заменить результирующим (суммарным) изгибающим моментом

который вызывает прямой изгиб в плоскости его действия относительно нейтральной оси n-n, перпендикулярной вект= ору Мизг. Эпюра суммарного момента имеет пространственное очертание и поэтому неудобна для построения и анализа. Поскольку все направления у круга с точки зрения прочности равноценны, то обычно эпюру Мизг спрямляют, помещая все ординаты в одну (например, вертикальную) плоскость. Обратим внимание на то, что центральный участок этой эпюры является нелинейным.

3. Строится эпюра крутящего момен= та Мz.

Наибольшие напряжения изгиба возника= ют в точках k и k', наиболее удаленных от нейтральной оси (рис. 3),

где Wизг - момент сопротивления при изгибе.

В этих же точках имеют место и наибольшие касательные напряжения кручения

где Wр - момент сопротивления при кручении.

Как следует из рис. 3, напряженное состояние является упрощенным = плоским (сочетание одноосного растяжения и чистого сдвига). Если вал выполнен из пластичного материала, оценка его прочности должна быть произведена по одно= му из критериев текучести. Например, по критерию Треска-Сен-Венана имеем

Учитывая, что Wр=3D2Wизг, для эквивалентных напряжений получаем

где - эквивалентный момент, с введением которого задача расчета вала = на совместное действие изгиба и кручения, сводится к расчету на эквивалентный изгиб.

Аналогично для Мэкв по критерию Губера-Мизеса получаем

Тогда условие прочности для вала из пласт= ичного материала будет иметь вид

Для стержня из хрупкого материала условие прочности следует записать в виде

где Мэкв должен бы= ть записан применительно к одному из критериев хрупкого разрушения. Например, = по критерию Мора

где m =3D [sp] / [sc].

Обратим внимание на особенности расч= ета при сочетании изгиба, растяжения и кручения стержня прямоугольного поперечн= ого сечения (рис. 4). Для выявления опасной точки здесь должны быть сравнены напряжения косого изгиба с растяжением в точке А, с эквивалентными напряжениями в точках В и С.

Полученные соотношения приобретают крайнюю необходимость и востребованность при выполнении Вами курсового прое= кта по основам конструирования при расчете на прочность и жесткость валов перед= ач.

 

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